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Theorem catciso 17359
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 17124 . . . . 5 Rel (𝑋 Func 𝑌)
2 catciso.b . . . . . . . . . . . . . 14 𝐵 = (Base‘𝐶)
3 eqid 2819 . . . . . . . . . . . . . 14 (Inv‘𝐶) = (Inv‘𝐶)
4 catciso.u . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
5 catciso.c . . . . . . . . . . . . . . . 16 𝐶 = (CatCat‘𝑈)
65catccat 17356 . . . . . . . . . . . . . . 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 17027 . . . . . . . . . . . . 13 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
1211eleq2d 2896 . . . . . . . . . . . 12 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
1312biimpa 479 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))
147adantr 483 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ Cat)
158adantr 483 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑋𝐵)
169adantr 483 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑌𝐵)
172, 3, 14, 15, 16invfun 17026 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → Fun (𝑋(Inv‘𝐶)𝑌))
18 funfvbrb 6814 . . . . . . . . . . . 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 234 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
21 eqid 2819 . . . . . . . . . . 11 (Sect‘𝐶) = (Sect‘𝐶)
222, 3, 14, 15, 16, 21isinv 17022 . . . . . . . . . 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 234 . . . . . . . . 9 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
2423simpld 497 . . . . . . . 8 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
25 eqid 2819 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
26 eqid 2819 . . . . . . . . 9 (comp‘𝐶) = (comp‘𝐶)
27 eqid 2819 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
282, 25, 26, 27, 21, 14, 15, 16issect 17015 . . . . . . . 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 234 . . . . . . 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 1136 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
315, 2, 4, 25, 8, 9catchom 17351 . . . . . . 7 (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
3231adantr 483 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
3330, 32eleqtrd 2913 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ (𝑋 Func 𝑌))
34 1st2nd 7730 . . . . 5 ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
351, 33, 34sylancr 589 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
36 1st2ndbr 7733 . . . . . . 7 ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → (1st𝐹)(𝑋 Func 𝑌)(2nd𝐹))
371, 33, 36sylancr 589 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹)(𝑋 Func 𝑌)(2nd𝐹))
38 catciso.r . . . . . . . . 9 𝑅 = (Base‘𝑋)
39 eqid 2819 . . . . . . . . 9 (Hom ‘𝑋) = (Hom ‘𝑋)
40 eqid 2819 . . . . . . . . 9 (Hom ‘𝑌) = (Hom ‘𝑌)
4137adantr 483 . . . . . . . . 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 17130 . . . . . . . 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 17124 . . . . . . . . . . . 12 Rel (𝑌 Func 𝑋)
4729simp2d 1137 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
485, 2, 4, 25, 9, 8catchom 17351 . . . . . . . . . . . . . 14 (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
4948adantr 483 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
5047, 49eleqtrd 2913 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋))
51 1st2ndbr 7733 . . . . . . . . . . . 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 589 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
5352adantr 483 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
5438, 45, 41funcf1 17128 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st𝐹):𝑅𝑆)
5554, 42ffvelrnd 6845 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st𝐹)‘𝑥) ∈ 𝑆)
5654, 43ffvelrnd 6845 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st𝐹)‘𝑦) ∈ 𝑆)
5745, 40, 39, 53, 55, 56funcf2 17130 . . . . . . . . 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 1138 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
594adantr 483 . . . . . . . . . . . . . . . . 17 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑈𝑉)
605, 2, 59, 26, 15, 16, 15, 33, 50catcco 17353 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))
61 eqid 2819 . . . . . . . . . . . . . . . . . 18 (idfunc𝑋) = (idfunc𝑋)
625, 2, 27, 61, 4, 8catcid 17355 . . . . . . . . . . . . . . . . 17 (𝜑 → ((Id‘𝐶)‘𝑋) = (idfunc𝑋))
6362adantr 483 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑋) = (idfunc𝑋))
6458, 60, 633eqtr3d 2862 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹) = (idfunc𝑋))
6564adantr 483 . . . . . . . . . . . . . 14 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹) = (idfunc𝑋))
6665fveq2d 6667 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (1st ‘(idfunc𝑋)))
6766fveq1d 6665 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑥) = ((1st ‘(idfunc𝑋))‘𝑥))
6833adantr 483 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝐹 ∈ (𝑋 Func 𝑌))
6950adantr 483 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋))
7038, 68, 69, 42cofu1 17146 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑥) = ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥)))
715, 2, 4catcbas 17349 . . . . . . . . . . . . . . . 16 (𝜑𝐵 = (𝑈 ∩ Cat))
72 inss2 4204 . . . . . . . . . . . . . . . 16 (𝑈 ∩ Cat) ⊆ Cat
7371, 72eqsstrdi 4019 . . . . . . . . . . . . . . 15 (𝜑𝐵 ⊆ Cat)
7473, 8sseldd 3966 . . . . . . . . . . . . . 14 (𝜑𝑋 ∈ Cat)
7574ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑋 ∈ Cat)
7661, 38, 75, 42idfu1 17142 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(idfunc𝑋))‘𝑥) = 𝑥)
7767, 70, 763eqtr3d 2862 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥)) = 𝑥)
7866fveq1d 6665 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑦) = ((1st ‘(idfunc𝑋))‘𝑦))
7938, 68, 69, 43cofu1 17146 . . . . . . . . . . . 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 17142 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(idfunc𝑋))‘𝑦) = 𝑦)
8178, 79, 803eqtr3d 2862 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑦)) = 𝑦)
8277, 81oveq12d 7166 . . . . . . . . . 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 6494 . . . . . . . . 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 234 . . . . . . . 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 6667 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (2nd ‘(idfunc𝑋)))
8685oveqd 7165 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))𝑦) = (𝑥(2nd ‘(idfunc𝑋))𝑦))
8738, 68, 69, 42, 43cofu2nd 17147 . . . . . . . . 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 17139 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd ‘(idfunc𝑋))𝑦) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
8986, 87, 883eqtr3d 2862 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
9023simprd 498 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
912, 25, 26, 27, 21, 14, 16, 15issect 17015 . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . 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 1138 . . . . . . . . . . . . 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 17353 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
95 eqid 2819 . . . . . . . . . . . . . . 15 (idfunc𝑌) = (idfunc𝑌)
965, 2, 27, 95, 4, 9catcid 17355 . . . . . . . . . . . . . 14 (𝜑 → ((Id‘𝐶)‘𝑌) = (idfunc𝑌))
9796adantr 483 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑌) = (idfunc𝑌))
9893, 94, 973eqtr3d 2862 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (idfunc𝑌))
9998adantr 483 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (idfunc𝑌))
10099fveq2d 6667 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (2nd ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (2nd ‘(idfunc𝑌)))
101100oveqd 7165 . . . . . . . . 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 17147 . . . . . . . . . 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 7166 . . . . . . . . . . 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 5725 . . . . . . . . . 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 2854 . . . . . . . . 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 3966 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑌 ∈ Cat)
10995, 45, 108, 40, 55, 56idfu2nd 17139 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((1st𝐹)‘𝑥)(2nd ‘(idfunc𝑌))((1st𝐹)‘𝑦)) = ( I ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦))))
110101, 105, 1093eqtr3d 2862 . . . . . . . 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 7042 . . . . . . 7 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦)))
112111ralrimivva 3189 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ∀𝑥𝑅𝑦𝑅 (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦)))
11338, 39, 40isffth2 17178 . . . . . 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 585 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹))
115 df-br 5058 . . . . 5 ((1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹) ↔ ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
116114, 115sylib 220 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
11735, 116eqeltrd 2911 . . 3 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
11838, 45, 37funcf1 17128 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹):𝑅𝑆)
11945, 38, 52funcf1 17128 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)):𝑆𝑅)
12064fveq2d 6667 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (1st ‘(idfunc𝑋)))
12138, 33, 50cofu1st 17145 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∘ (1st𝐹)))
12274adantr 483 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ Cat)
12361, 38, 122idfu1st 17141 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(idfunc𝑋)) = ( I ↾ 𝑅))
124120, 121, 1233eqtr3d 2862 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∘ (1st𝐹)) = ( I ↾ 𝑅))
12598fveq2d 6667 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (1st ‘(idfunc𝑌)))
12645, 50, 33cofu1st 17145 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ((1st𝐹) ∘ (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
12773, 9sseldd 3966 . . . . . . 7 (𝜑𝑌 ∈ Cat)
128127adantr 483 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ Cat)
12995, 45, 128idfu1st 17141 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(idfunc𝑌)) = ( I ↾ 𝑆))
130125, 126, 1293eqtr3d 2862 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((1st𝐹) ∘ (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ( I ↾ 𝑆))
131118, 119, 124, 130fcof1od 7042 . . 3 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹):𝑅1-1-onto𝑆)
132117, 131jca 514 . 2 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆))
1337adantr 483 . . 3 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐶 ∈ Cat)
1348adantr 483 . . 3 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝑋𝐵)
1359adantr 483 . . 3 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝑌𝐵)
136 inss1 4203 . . . . . . 7 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Full 𝑌)
137 fullfunc 17168 . . . . . . 7 (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
138136, 137sstri 3974 . . . . . 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, 139sseldi 3963 . . . . 5 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 ∈ (𝑋 Func 𝑌))
1411, 140, 34sylancr 589 . . . 4 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
1424adantr 483 . . . . 5 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝑈𝑉)
143 eqid 2819 . . . . 5 (𝑥𝑆, 𝑦𝑆(((1st𝐹)‘𝑥)(2nd𝐹)((1st𝐹)‘𝑦))) = (𝑥𝑆, 𝑦𝑆(((1st𝐹)‘𝑥)(2nd𝐹)((1st𝐹)‘𝑦)))
144141, 139eqeltrrd 2912 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
145144, 115sylibr 236 . . . . 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 17358 . . . 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 5079 . . 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 17028 . 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 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  cin 3933  wss 3934  cop 4565   class class class wbr 5057   I cid 5452  ccnv 5547  dom cdm 5548  cres 5550  ccom 5552  Rel wrel 5553  Fun wfun 6342  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7148  cmpo 7150  1st c1st 7679  2nd c2nd 7680  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  Idccid 16928  Sectcsect 17006  Invcinv 17007  Isociso 17008   Func cfunc 17116  idfunccidfu 17117  func ccofu 17118   Full cful 17164   Faith cfth 17165  CatCatccatc 17346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12885  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-hom 16581  df-cco 16582  df-cat 16931  df-cid 16932  df-sect 17009  df-inv 17010  df-iso 17011  df-func 17120  df-idfu 17121  df-cofu 17122  df-full 17166  df-fth 17167  df-catc 17347
This theorem is referenced by:  yoniso  17527
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