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Theorem catciso 17997
Description: A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. Remark 3.28(2) in [Adamek] p. 34. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
catciso.c 𝐶 = (CatCat‘𝑈)
catciso.b 𝐵 = (Base‘𝐶)
catciso.r 𝑅 = (Base‘𝑋)
catciso.s 𝑆 = (Base‘𝑌)
catciso.u (𝜑𝑈𝑉)
catciso.x (𝜑𝑋𝐵)
catciso.y (𝜑𝑌𝐵)
catciso.i 𝐼 = (Iso‘𝐶)
Assertion
Ref Expression
catciso (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)))

Proof of Theorem catciso
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 17748 . . . . 5 Rel (𝑋 Func 𝑌)
2 catciso.b . . . . . . . . . . . . . 14 𝐵 = (Base‘𝐶)
3 eqid 2736 . . . . . . . . . . . . . 14 (Inv‘𝐶) = (Inv‘𝐶)
4 catciso.u . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
5 catciso.c . . . . . . . . . . . . . . . 16 𝐶 = (CatCat‘𝑈)
65catccat 17994 . . . . . . . . . . . . . . 15 (𝑈𝑉𝐶 ∈ Cat)
74, 6syl 17 . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ Cat)
8 catciso.x . . . . . . . . . . . . . 14 (𝜑𝑋𝐵)
9 catciso.y . . . . . . . . . . . . . 14 (𝜑𝑌𝐵)
10 catciso.i . . . . . . . . . . . . . 14 𝐼 = (Iso‘𝐶)
112, 3, 7, 8, 9, 10isoval 17648 . . . . . . . . . . . . 13 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
1211eleq2d 2823 . . . . . . . . . . . 12 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
1312biimpa 477 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))
147adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ Cat)
158adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑋𝐵)
169adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑌𝐵)
172, 3, 14, 15, 16invfun 17647 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → Fun (𝑋(Inv‘𝐶)𝑌))
18 funfvbrb 7001 . . . . . . . . . . . 12 (Fun (𝑋(Inv‘𝐶)𝑌) → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
1917, 18syl 17 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
2013, 19mpbid 231 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
21 eqid 2736 . . . . . . . . . . 11 (Sect‘𝐶) = (Sect‘𝐶)
222, 3, 14, 15, 16, 21isinv 17643 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
2320, 22mpbid 231 . . . . . . . . 9 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
2423simpld 495 . . . . . . . 8 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
25 eqid 2736 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
26 eqid 2736 . . . . . . . . 9 (comp‘𝐶) = (comp‘𝐶)
27 eqid 2736 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
282, 25, 26, 27, 21, 14, 15, 16issect 17636 . . . . . . . 8 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
2924, 28mpbid 231 . . . . . . 7 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
3029simp1d 1142 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
315, 2, 4, 25, 8, 9catchom 17989 . . . . . . 7 (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
3231adantr 481 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
3330, 32eleqtrd 2840 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ (𝑋 Func 𝑌))
34 1st2nd 7971 . . . . 5 ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
351, 33, 34sylancr 587 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
36 1st2ndbr 7974 . . . . . . 7 ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → (1st𝐹)(𝑋 Func 𝑌)(2nd𝐹))
371, 33, 36sylancr 587 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹)(𝑋 Func 𝑌)(2nd𝐹))
38 catciso.r . . . . . . . . 9 𝑅 = (Base‘𝑋)
39 eqid 2736 . . . . . . . . 9 (Hom ‘𝑋) = (Hom ‘𝑋)
40 eqid 2736 . . . . . . . . 9 (Hom ‘𝑌) = (Hom ‘𝑌)
4137adantr 481 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st𝐹)(𝑋 Func 𝑌)(2nd𝐹))
42 simprl 769 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥𝑅)
43 simprr 771 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦𝑅)
4438, 39, 40, 41, 42, 43funcf2 17754 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦)))
45 catciso.s . . . . . . . . . 10 𝑆 = (Base‘𝑌)
46 relfunc 17748 . . . . . . . . . . . 12 Rel (𝑌 Func 𝑋)
4729simp2d 1143 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
485, 2, 4, 25, 9, 8catchom 17989 . . . . . . . . . . . . . 14 (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
4948adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
5047, 49eleqtrd 2840 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋))
51 1st2ndbr 7974 . . . . . . . . . . . 12 ((Rel (𝑌 Func 𝑋) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
5246, 50, 51sylancr 587 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
5352adantr 481 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
5438, 45, 41funcf1 17752 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st𝐹):𝑅𝑆)
5554, 42ffvelcdmd 7036 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st𝐹)‘𝑥) ∈ 𝑆)
5654, 43ffvelcdmd 7036 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st𝐹)‘𝑦) ∈ 𝑆)
5745, 40, 39, 53, 55, 56funcf2 17754 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦)):(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦))⟶(((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥))(Hom ‘𝑋)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑦))))
5829simp3d 1144 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
594adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑈𝑉)
605, 2, 59, 26, 15, 16, 15, 33, 50catcco 17991 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))
61 eqid 2736 . . . . . . . . . . . . . . . . . 18 (idfunc𝑋) = (idfunc𝑋)
625, 2, 27, 61, 4, 8catcid 17993 . . . . . . . . . . . . . . . . 17 (𝜑 → ((Id‘𝐶)‘𝑋) = (idfunc𝑋))
6362adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑋) = (idfunc𝑋))
6458, 60, 633eqtr3d 2784 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹) = (idfunc𝑋))
6564adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹) = (idfunc𝑋))
6665fveq2d 6846 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (1st ‘(idfunc𝑋)))
6766fveq1d 6844 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑥) = ((1st ‘(idfunc𝑋))‘𝑥))
6833adantr 481 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝐹 ∈ (𝑋 Func 𝑌))
6950adantr 481 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋))
7038, 68, 69, 42cofu1 17770 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑥) = ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥)))
715, 2, 4catcbas 17987 . . . . . . . . . . . . . . . 16 (𝜑𝐵 = (𝑈 ∩ Cat))
72 inss2 4189 . . . . . . . . . . . . . . . 16 (𝑈 ∩ Cat) ⊆ Cat
7371, 72eqsstrdi 3998 . . . . . . . . . . . . . . 15 (𝜑𝐵 ⊆ Cat)
7473, 8sseldd 3945 . . . . . . . . . . . . . 14 (𝜑𝑋 ∈ Cat)
7574ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑋 ∈ Cat)
7661, 38, 75, 42idfu1 17766 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(idfunc𝑋))‘𝑥) = 𝑥)
7767, 70, 763eqtr3d 2784 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥)) = 𝑥)
7866fveq1d 6844 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑦) = ((1st ‘(idfunc𝑋))‘𝑦))
7938, 68, 69, 43cofu1 17770 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑦) = ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑦)))
8061, 38, 75, 43idfu1 17766 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(idfunc𝑋))‘𝑦) = 𝑦)
8178, 79, 803eqtr3d 2784 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑦)) = 𝑦)
8277, 81oveq12d 7375 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥))(Hom ‘𝑋)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑦))) = (𝑥(Hom ‘𝑋)𝑦))
8382feq3d 6655 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦)):(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦))⟶(((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥))(Hom ‘𝑋)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑦))) ↔ (((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦)):(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦))⟶(𝑥(Hom ‘𝑋)𝑦)))
8457, 83mpbid 231 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦)):(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦))⟶(𝑥(Hom ‘𝑋)𝑦))
8565fveq2d 6846 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (2nd ‘(idfunc𝑋)))
8685oveqd 7374 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))𝑦) = (𝑥(2nd ‘(idfunc𝑋))𝑦))
8738, 68, 69, 42, 43cofu2nd 17771 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))𝑦) = ((((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))
8861, 38, 75, 39, 42, 43idfu2nd 17763 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd ‘(idfunc𝑋))𝑦) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
8986, 87, 883eqtr3d 2784 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
9023simprd 496 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
912, 25, 26, 27, 21, 14, 16, 15issect 17636 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = ((Id‘𝐶)‘𝑌))))
9290, 91mpbid 231 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = ((Id‘𝐶)‘𝑌)))
9392simp3d 1144 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = ((Id‘𝐶)‘𝑌))
945, 2, 59, 26, 16, 15, 16, 50, 33catcco 17991 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
95 eqid 2736 . . . . . . . . . . . . . . 15 (idfunc𝑌) = (idfunc𝑌)
965, 2, 27, 95, 4, 9catcid 17993 . . . . . . . . . . . . . 14 (𝜑 → ((Id‘𝐶)‘𝑌) = (idfunc𝑌))
9796adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑌) = (idfunc𝑌))
9893, 94, 973eqtr3d 2784 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (idfunc𝑌))
9998adantr 481 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (idfunc𝑌))
10099fveq2d 6846 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (2nd ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (2nd ‘(idfunc𝑌)))
101100oveqd 7374 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((1st𝐹)‘𝑥)(2nd ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))((1st𝐹)‘𝑦)) = (((1st𝐹)‘𝑥)(2nd ‘(idfunc𝑌))((1st𝐹)‘𝑦)))
10245, 69, 68, 55, 56cofu2nd 17771 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((1st𝐹)‘𝑥)(2nd ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))((1st𝐹)‘𝑦)) = ((((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥))(2nd𝐹)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑦))) ∘ (((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦))))
10377, 81oveq12d 7375 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥))(2nd𝐹)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑦))) = (𝑥(2nd𝐹)𝑦))
104103coeq1d 5817 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥))(2nd𝐹)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑦))) ∘ (((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦))) = ((𝑥(2nd𝐹)𝑦) ∘ (((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦))))
105102, 104eqtrd 2776 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((1st𝐹)‘𝑥)(2nd ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))((1st𝐹)‘𝑦)) = ((𝑥(2nd𝐹)𝑦) ∘ (((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦))))
10673ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝐵 ⊆ Cat)
1079ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑌𝐵)
108106, 107sseldd 3945 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑌 ∈ Cat)
10995, 45, 108, 40, 55, 56idfu2nd 17763 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((1st𝐹)‘𝑥)(2nd ‘(idfunc𝑌))((1st𝐹)‘𝑦)) = ( I ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦))))
110101, 105, 1093eqtr3d 2784 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((𝑥(2nd𝐹)𝑦) ∘ (((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦))) = ( I ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦))))
11144, 84, 89, 110fcof1od 7240 . . . . . . 7 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦)))
112111ralrimivva 3197 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ∀𝑥𝑅𝑦𝑅 (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦)))
11338, 39, 40isffth2 17803 . . . . . 6 ((1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹) ↔ ((1st𝐹)(𝑋 Func 𝑌)(2nd𝐹) ∧ ∀𝑥𝑅𝑦𝑅 (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦))))
11437, 112, 113sylanbrc 583 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹))
115 df-br 5106 . . . . 5 ((1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹) ↔ ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
116114, 115sylib 217 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
11735, 116eqeltrd 2838 . . 3 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
11838, 45, 37funcf1 17752 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹):𝑅𝑆)
11945, 38, 52funcf1 17752 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)):𝑆𝑅)
12064fveq2d 6846 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (1st ‘(idfunc𝑋)))
12138, 33, 50cofu1st 17769 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∘ (1st𝐹)))
12274adantr 481 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ Cat)
12361, 38, 122idfu1st 17765 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(idfunc𝑋)) = ( I ↾ 𝑅))
124120, 121, 1233eqtr3d 2784 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∘ (1st𝐹)) = ( I ↾ 𝑅))
12598fveq2d 6846 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (1st ‘(idfunc𝑌)))
12645, 50, 33cofu1st 17769 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ((1st𝐹) ∘ (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
12773, 9sseldd 3945 . . . . . . 7 (𝜑𝑌 ∈ Cat)
128127adantr 481 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ Cat)
12995, 45, 128idfu1st 17765 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(idfunc𝑌)) = ( I ↾ 𝑆))
130125, 126, 1293eqtr3d 2784 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((1st𝐹) ∘ (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ( I ↾ 𝑆))
131118, 119, 124, 130fcof1od 7240 . . 3 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹):𝑅1-1-onto𝑆)
132117, 131jca 512 . 2 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆))
1337adantr 481 . . 3 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐶 ∈ Cat)
1348adantr 481 . . 3 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝑋𝐵)
1359adantr 481 . . 3 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝑌𝐵)
136 inss1 4188 . . . . . . 7 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Full 𝑌)
137 fullfunc 17793 . . . . . . 7 (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
138136, 137sstri 3953 . . . . . 6 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Func 𝑌)
139 simprl 769 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
140138, 139sselid 3942 . . . . 5 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 ∈ (𝑋 Func 𝑌))
1411, 140, 34sylancr 587 . . . 4 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
1424adantr 481 . . . . 5 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝑈𝑉)
143 eqid 2736 . . . . 5 (𝑥𝑆, 𝑦𝑆(((1st𝐹)‘𝑥)(2nd𝐹)((1st𝐹)‘𝑦))) = (𝑥𝑆, 𝑦𝑆(((1st𝐹)‘𝑥)(2nd𝐹)((1st𝐹)‘𝑦)))
144141, 139eqeltrrd 2839 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
145144, 115sylibr 233 . . . . 5 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → (1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹))
146 simprr 771 . . . . 5 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → (1st𝐹):𝑅1-1-onto𝑆)
1475, 2, 38, 45, 142, 134, 135, 3, 143, 145, 146catcisolem 17996 . . . 4 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → ⟨(1st𝐹), (2nd𝐹)⟩(𝑋(Inv‘𝐶)𝑌)⟨(1st𝐹), (𝑥𝑆, 𝑦𝑆(((1st𝐹)‘𝑥)(2nd𝐹)((1st𝐹)‘𝑦)))⟩)
148141, 147eqbrtrd 5127 . . 3 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹(𝑋(Inv‘𝐶)𝑌)⟨(1st𝐹), (𝑥𝑆, 𝑦𝑆(((1st𝐹)‘𝑥)(2nd𝐹)((1st𝐹)‘𝑦)))⟩)
1492, 3, 133, 134, 135, 10, 148inviso1 17649 . 2 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 ∈ (𝑋𝐼𝑌))
150132, 149impbida 799 1 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  cin 3909  wss 3910  cop 4592   class class class wbr 5105   I cid 5530  ccnv 5632  dom cdm 5633  cres 5635  ccom 5637  Rel wrel 5638  Fun wfun 6490  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Idccid 17545  Sectcsect 17627  Invcinv 17628  Isociso 17629   Func cfunc 17740  idfunccidfu 17741  func ccofu 17742   Full cful 17789   Faith cfth 17790  CatCatccatc 17984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-sect 17630  df-inv 17631  df-iso 17632  df-func 17744  df-idfu 17745  df-cofu 17746  df-full 17791  df-fth 17792  df-catc 17985
This theorem is referenced by:  yoniso  18174  thincciso  47059
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