:: ENS_1 semantic presentation
:: deftheorem Def1 defines Funcs ENS_1:def 1 :
theorem Th1: :: ENS_1:1
theorem Th2: :: ENS_1:2
theorem Th3: :: ENS_1:3
definition
let V be non
empty set ;
func Maps c1 -> set equals :: ENS_1:def 2
{ [[A,B],f] where A is Element of V, B is Element of V, f is Element of Funcs V : ( ( B = {} implies A = {} ) & f is Function of A,B ) } ;
coherence
{ [[A,B],f] where A is Element of V, B is Element of V, f is Element of Funcs V : ( ( B = {} implies A = {} ) & f is Function of A,B ) } is set
;
end;
:: deftheorem Def2 defines Maps ENS_1:def 2 :
theorem Th4: :: ENS_1:4
theorem Th5: :: ENS_1:5
theorem Th6: :: ENS_1:6
theorem Th7: :: ENS_1:7
Lemma34:
for x1, y1, x2, y2, x3, y3 being set st [[x1,x2],x3] = [[y1,y2],y3] holds
( x1 = y1 & x2 = y2 )
:: deftheorem Def3 ENS_1:def 3 :
canceled;
:: deftheorem Def4 defines dom ENS_1:def 4 :
:: deftheorem Def5 defines cod ENS_1:def 5 :
theorem Th8: :: ENS_1:8
Lemma42:
for V being non empty set
for m, m' being Element of Maps V st m `2 = m' `2 & dom m = dom m' & cod m = cod m' holds
m = m'
theorem Th9: :: ENS_1:9
Lemma44:
for V being non empty set
for m being Element of Maps V holds
( dom (m `2 ) = dom m & rng (m `2 ) c= cod m )
theorem Th10: :: ENS_1:10
:: deftheorem Def6 defines id$ ENS_1:def 6 :
theorem Th11: :: ENS_1:11
:: deftheorem Def7 defines * ENS_1:def 7 :
theorem Th12: :: ENS_1:12
theorem Th13: :: ENS_1:13
theorem Th14: :: ENS_1:14
:: deftheorem Def8 defines Maps ENS_1:def 8 :
theorem Th15: :: ENS_1:15
theorem Th16: :: ENS_1:16
theorem Th17: :: ENS_1:17
Lemma57:
for V being non empty set
for A, B being Element of V
for f being Function st [[A,B],f] in Maps A,B holds
( ( B = {} implies A = {} ) & f is Function of A,B )
theorem Th18: :: ENS_1:18
theorem Th19: :: ENS_1:19
theorem Th20: :: ENS_1:20
Lemma61:
for V being non empty set
for W being non empty Subset of V
for A, B being Element of W
for A', B' being Element of V st A = A' & B = B' holds
Maps A,B = Maps A',B'
:: deftheorem Def9 defines surjective ENS_1:def 9 :
definition
let V be non
empty set ;
func fDom c1 -> Function of
Maps a1,
a1 means :
Def10:
:: ENS_1:def 10
for
m being
Element of
Maps V holds
it . m = dom m;
existence
ex b1 being Function of Maps V,V st
for m being Element of Maps V holds b1 . m = dom m
uniqueness
for b1, b2 being Function of Maps V,V st ( for m being Element of Maps V holds b1 . m = dom m ) & ( for m being Element of Maps V holds b2 . m = dom m ) holds
b1 = b2
func fCod c1 -> Function of
Maps a1,
a1 means :
Def11:
:: ENS_1:def 11
for
m being
Element of
Maps V holds
it . m = cod m;
existence
ex b1 being Function of Maps V,V st
for m being Element of Maps V holds b1 . m = cod m
uniqueness
for b1, b2 being Function of Maps V,V st ( for m being Element of Maps V holds b1 . m = cod m ) & ( for m being Element of Maps V holds b2 . m = cod m ) holds
b1 = b2
func fComp c1 -> PartFunc of
[:(Maps a1),(Maps a1):],
Maps a1 means :
Def12:
:: ENS_1:def 12
( ( for
m2,
m1 being
Element of
Maps V holds
(
[m2,m1] in dom it iff
dom m2 = cod m1 ) ) & ( for
m2,
m1 being
Element of
Maps V st
dom m2 = cod m1 holds
it . [m2,m1] = m2 * m1 ) );
existence
ex b1 being PartFunc of [:(Maps V),(Maps V):], Maps V st
( ( for m2, m1 being Element of Maps V holds
( [m2,m1] in dom b1 iff dom m2 = cod m1 ) ) & ( for m2, m1 being Element of Maps V st dom m2 = cod m1 holds
b1 . [m2,m1] = m2 * m1 ) )
uniqueness
for b1, b2 being PartFunc of [:(Maps V),(Maps V):], Maps V st ( for m2, m1 being Element of Maps V holds
( [m2,m1] in dom b1 iff dom m2 = cod m1 ) ) & ( for m2, m1 being Element of Maps V st dom m2 = cod m1 holds
b1 . [m2,m1] = m2 * m1 ) & ( for m2, m1 being Element of Maps V holds
( [m2,m1] in dom b2 iff dom m2 = cod m1 ) ) & ( for m2, m1 being Element of Maps V st dom m2 = cod m1 holds
b2 . [m2,m1] = m2 * m1 ) holds
b1 = b2
func fId c1 -> Function of
a1,
Maps a1 means :
Def13:
:: ENS_1:def 13
for
A being
Element of
V holds
it . A = id$ A;
existence
ex b1 being Function of V, Maps V st
for A being Element of V holds b1 . A = id$ A
uniqueness
for b1, b2 being Function of V, Maps V st ( for A being Element of V holds b1 . A = id$ A ) & ( for A being Element of V holds b2 . A = id$ A ) holds
b1 = b2
end;
:: deftheorem Def10 defines fDom ENS_1:def 10 :
:: deftheorem Def11 defines fCod ENS_1:def 11 :
:: deftheorem Def12 defines fComp ENS_1:def 12 :
:: deftheorem Def13 defines fId ENS_1:def 13 :
definition
let V be non
empty set ;
func Ens c1 -> CatStr equals :: ENS_1:def 14
CatStr(#
V,
(Maps V),
(fDom V),
(fCod V),
(fComp V),
(fId V) #);
coherence
CatStr(# V,(Maps V),(fDom V),(fCod V),(fComp V),(fId V) #) is CatStr
;
end;
:: deftheorem Def14 defines Ens ENS_1:def 14 :
theorem Th21: :: ENS_1:21
theorem Th22: :: ENS_1:22
:: deftheorem Def15 defines @ ENS_1:def 15 :
theorem Th23: :: ENS_1:23
:: deftheorem Def16 defines @ ENS_1:def 16 :
theorem Th24: :: ENS_1:24
:: deftheorem Def17 defines @ ENS_1:def 17 :
theorem Th25: :: ENS_1:25
:: deftheorem Def18 defines @ ENS_1:def 18 :
theorem Th26: :: ENS_1:26
theorem Th27: :: ENS_1:27
Lemma103:
for V being non empty set
for a, b being Object of (Ens V) st Hom a,b <> {} holds
Funcs (@ a),(@ b) <> {}
theorem Th28: :: ENS_1:28
theorem Th29: :: ENS_1:29
theorem Th30: :: ENS_1:30
theorem Th31: :: ENS_1:31
theorem Th32: :: ENS_1:32
theorem Th33: :: ENS_1:33
theorem Th34: :: ENS_1:34
theorem Th35: :: ENS_1:35
theorem Th36: :: ENS_1:36
theorem Th37: :: ENS_1:37
theorem Th38: :: ENS_1:38
theorem Th39: :: ENS_1:39
theorem Th40: :: ENS_1:40
:: deftheorem Def19 defines Hom ENS_1:def 19 :
theorem Th41: :: ENS_1:41
theorem Th42: :: ENS_1:42
definition
let C be
Category;
let a be
Object of
C;
let f be
Morphism of
C;
func hom c2,
c3 -> Function of
Hom a2,
(dom a3),
Hom a2,
(cod a3) means :
Def20:
:: ENS_1:def 20
for
g being
Morphism of
C st
g in Hom a,
(dom f) holds
it . g = f * g;
existence
ex b1 being Function of Hom a,(dom f), Hom a,(cod f) st
for g being Morphism of C st g in Hom a,(dom f) holds
b1 . g = f * g
uniqueness
for b1, b2 being Function of Hom a,(dom f), Hom a,(cod f) st ( for g being Morphism of C st g in Hom a,(dom f) holds
b1 . g = f * g ) & ( for g being Morphism of C st g in Hom a,(dom f) holds
b2 . g = f * g ) holds
b1 = b2
func hom c3,
c2 -> Function of
Hom (cod a3),
a2,
Hom (dom a3),
a2 means :
Def21:
:: ENS_1:def 21
for
g being
Morphism of
C st
g in Hom (cod f),
a holds
it . g = g * f;
existence
ex b1 being Function of Hom (cod f),a, Hom (dom f),a st
for g being Morphism of C st g in Hom (cod f),a holds
b1 . g = g * f
uniqueness
for b1, b2 being Function of Hom (cod f),a, Hom (dom f),a st ( for g being Morphism of C st g in Hom (cod f),a holds
b1 . g = g * f ) & ( for g being Morphism of C st g in Hom (cod f),a holds
b2 . g = g * f ) holds
b1 = b2
end;
:: deftheorem Def20 defines hom ENS_1:def 20 :
:: deftheorem Def21 defines hom ENS_1:def 21 :
theorem Th43: :: ENS_1:43
theorem Th44: :: ENS_1:44
theorem Th45: :: ENS_1:45
theorem Th46: :: ENS_1:46
theorem Th47: :: ENS_1:47
theorem Th48: :: ENS_1:48
definition
let C be
Category;
let a be
Object of
C;
func hom?- c2 -> Function of the
Morphisms of
a1,
Maps (Hom a1) means :
Def22:
:: ENS_1:def 22
for
f being
Morphism of
C holds
it . f = [[(Hom a,(dom f)),(Hom a,(cod f))],(hom a,f)];
existence
ex b1 being Function of the Morphisms of C, Maps (Hom C) st
for f being Morphism of C holds b1 . f = [[(Hom a,(dom f)),(Hom a,(cod f))],(hom a,f)]
uniqueness
for b1, b2 being Function of the Morphisms of C, Maps (Hom C) st ( for f being Morphism of C holds b1 . f = [[(Hom a,(dom f)),(Hom a,(cod f))],(hom a,f)] ) & ( for f being Morphism of C holds b2 . f = [[(Hom a,(dom f)),(Hom a,(cod f))],(hom a,f)] ) holds
b1 = b2
func hom-? c2 -> Function of the
Morphisms of
a1,
Maps (Hom a1) means :
Def23:
:: ENS_1:def 23
for
f being
Morphism of
C holds
it . f = [[(Hom (cod f),a),(Hom (dom f),a)],(hom f,a)];
existence
ex b1 being Function of the Morphisms of C, Maps (Hom C) st
for f being Morphism of C holds b1 . f = [[(Hom (cod f),a),(Hom (dom f),a)],(hom f,a)]
uniqueness
for b1, b2 being Function of the Morphisms of C, Maps (Hom C) st ( for f being Morphism of C holds b1 . f = [[(Hom (cod f),a),(Hom (dom f),a)],(hom f,a)] ) & ( for f being Morphism of C holds b2 . f = [[(Hom (cod f),a),(Hom (dom f),a)],(hom f,a)] ) holds
b1 = b2
end;
:: deftheorem Def22 defines hom?- ENS_1:def 22 :
:: deftheorem Def23 defines hom-? ENS_1:def 23 :
Lemma139:
for V being non empty set
for C being Category
for T being Function of the Morphisms of C, Maps (Hom C) st Hom C c= V holds
T is Function of the Morphisms of C,the Morphisms of (Ens V)
Lemma141:
for V being non empty set
for C being Category
for a, c being Object of C st Hom C c= V holds
for d being Object of (Ens V) st d = Hom a,c holds
(hom?- a) . (id c) = id d
Lemma142:
for V being non empty set
for C being Category
for c, a being Object of C st Hom C c= V holds
for d being Object of (Ens V) st d = Hom c,a holds
(hom-? a) . (id c) = id d
theorem Th49: :: ENS_1:49
theorem Th50: :: ENS_1:50
theorem Th51: :: ENS_1:51
definition
let C be
Category;
let f be
Morphism of
C;
let g be
Morphism of
C;
func hom c2,
c3 -> Function of
Hom (cod a2),
(dom a3),
Hom (dom a2),
(cod a3) means :
Def24:
:: ENS_1:def 24
for
h being
Morphism of
C st
h in Hom (cod f),
(dom g) holds
it . h = (g * h) * f;
existence
ex b1 being Function of Hom (cod f),(dom g), Hom (dom f),(cod g) st
for h being Morphism of C st h in Hom (cod f),(dom g) holds
b1 . h = (g * h) * f
uniqueness
for b1, b2 being Function of Hom (cod f),(dom g), Hom (dom f),(cod g) st ( for h being Morphism of C st h in Hom (cod f),(dom g) holds
b1 . h = (g * h) * f ) & ( for h being Morphism of C st h in Hom (cod f),(dom g) holds
b2 . h = (g * h) * f ) holds
b1 = b2
end;
:: deftheorem Def24 defines hom ENS_1:def 24 :
theorem Th52: :: ENS_1:52
theorem Th53: :: ENS_1:53
theorem Th54: :: ENS_1:54
theorem Th55: :: ENS_1:55
theorem Th56: :: ENS_1:56
definition
let C be
Category;
func hom?? c1 -> Function of the
Morphisms of
[:a1,a1:],
Maps (Hom a1) means :
Def25:
:: ENS_1:def 25
for
f,
g being
Morphism of
C holds
it . [f,g] = [[(Hom (cod f),(dom g)),(Hom (dom f),(cod g))],(hom f,g)];
existence
ex b1 being Function of the Morphisms of [:C,C:], Maps (Hom C) st
for f, g being Morphism of C holds b1 . [f,g] = [[(Hom (cod f),(dom g)),(Hom (dom f),(cod g))],(hom f,g)]
uniqueness
for b1, b2 being Function of the Morphisms of [:C,C:], Maps (Hom C) st ( for f, g being Morphism of C holds b1 . [f,g] = [[(Hom (cod f),(dom g)),(Hom (dom f),(cod g))],(hom f,g)] ) & ( for f, g being Morphism of C holds b2 . [f,g] = [[(Hom (cod f),(dom g)),(Hom (dom f),(cod g))],(hom f,g)] ) holds
b1 = b2
end;
:: deftheorem Def25 defines hom?? ENS_1:def 25 :
theorem Th57: :: ENS_1:57
Lemma155:
for V being non empty set
for C being Category
for a, b being Object of C st Hom C c= V holds
for d being Object of (Ens V) st d = Hom a,b holds
(hom?? C) . [(id a),(id b)] = id d
theorem Th58: :: ENS_1:58
:: deftheorem Def26 defines hom?- ENS_1:def 26 :
:: deftheorem Def27 defines hom-? ENS_1:def 27 :
:: deftheorem Def28 defines hom?? ENS_1:def 28 :
theorem Th59: :: ENS_1:59
theorem Th60: :: ENS_1:60
theorem Th61: :: ENS_1:61
theorem Th62: :: ENS_1:62
theorem Th63: :: ENS_1:63
theorem Th64: :: ENS_1:64
theorem Th65: :: ENS_1:65
theorem Th66: :: ENS_1:66