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Theorem catciso 17022
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 16787 . . . . 5 Rel (𝑋 Func 𝑌)
2 catciso.b . . . . . . . . . . . . . 14 𝐵 = (Base‘𝐶)
3 eqid 2765 . . . . . . . . . . . . . 14 (Inv‘𝐶) = (Inv‘𝐶)
4 catciso.u . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
5 catciso.c . . . . . . . . . . . . . . . 16 𝐶 = (CatCat‘𝑈)
65catccat 17019 . . . . . . . . . . . . . . 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 16690 . . . . . . . . . . . . 13 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
1211eleq2d 2830 . . . . . . . . . . . 12 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
1312biimpa 468 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))
147adantr 472 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ Cat)
158adantr 472 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑋𝐵)
169adantr 472 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑌𝐵)
172, 3, 14, 15, 16invfun 16689 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → Fun (𝑋(Inv‘𝐶)𝑌))
18 funfvbrb 6520 . . . . . . . . . . . 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 223 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
21 eqid 2765 . . . . . . . . . . 11 (Sect‘𝐶) = (Sect‘𝐶)
222, 3, 14, 15, 16, 21isinv 16685 . . . . . . . . . 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 223 . . . . . . . . 9 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
2423simpld 488 . . . . . . . 8 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
25 eqid 2765 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
26 eqid 2765 . . . . . . . . 9 (comp‘𝐶) = (comp‘𝐶)
27 eqid 2765 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
282, 25, 26, 27, 21, 14, 15, 16issect 16678 . . . . . . . 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 223 . . . . . . 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 1172 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
315, 2, 4, 25, 8, 9catchom 17014 . . . . . . 7 (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
3231adantr 472 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
3330, 32eleqtrd 2846 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ (𝑋 Func 𝑌))
34 1st2nd 7414 . . . . 5 ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
351, 33, 34sylancr 581 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
36 1st2ndbr 7417 . . . . . . 7 ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → (1st𝐹)(𝑋 Func 𝑌)(2nd𝐹))
371, 33, 36sylancr 581 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹)(𝑋 Func 𝑌)(2nd𝐹))
38 catciso.r . . . . . . . . 9 𝑅 = (Base‘𝑋)
39 eqid 2765 . . . . . . . . 9 (Hom ‘𝑋) = (Hom ‘𝑋)
40 eqid 2765 . . . . . . . . 9 (Hom ‘𝑌) = (Hom ‘𝑌)
4137adantr 472 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st𝐹)(𝑋 Func 𝑌)(2nd𝐹))
42 simprl 787 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥𝑅)
43 simprr 789 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦𝑅)
4438, 39, 40, 41, 42, 43funcf2 16793 . . . . . . . 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 16787 . . . . . . . . . . . 12 Rel (𝑌 Func 𝑋)
4729simp2d 1173 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
485, 2, 4, 25, 9, 8catchom 17014 . . . . . . . . . . . . . 14 (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
4948adantr 472 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
5047, 49eleqtrd 2846 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋))
51 1st2ndbr 7417 . . . . . . . . . . . 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 581 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
5352adantr 472 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
5438, 45, 41funcf1 16791 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st𝐹):𝑅𝑆)
5554, 42ffvelrnd 6550 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st𝐹)‘𝑥) ∈ 𝑆)
5654, 43ffvelrnd 6550 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st𝐹)‘𝑦) ∈ 𝑆)
5745, 40, 39, 53, 55, 56funcf2 16793 . . . . . . . . 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 1174 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
594adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑈𝑉)
605, 2, 59, 26, 15, 16, 15, 33, 50catcco 17016 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))
61 eqid 2765 . . . . . . . . . . . . . . . . . 18 (idfunc𝑋) = (idfunc𝑋)
625, 2, 27, 61, 4, 8catcid 17018 . . . . . . . . . . . . . . . . 17 (𝜑 → ((Id‘𝐶)‘𝑋) = (idfunc𝑋))
6362adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑋) = (idfunc𝑋))
6458, 60, 633eqtr3d 2807 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹) = (idfunc𝑋))
6564adantr 472 . . . . . . . . . . . . . 14 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹) = (idfunc𝑋))
6665fveq2d 6379 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (1st ‘(idfunc𝑋)))
6766fveq1d 6377 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑥) = ((1st ‘(idfunc𝑋))‘𝑥))
6833adantr 472 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝐹 ∈ (𝑋 Func 𝑌))
6950adantr 472 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋))
7038, 68, 69, 42cofu1 16809 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑥) = ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥)))
715, 2, 4catcbas 17012 . . . . . . . . . . . . . . . 16 (𝜑𝐵 = (𝑈 ∩ Cat))
72 inss2 3993 . . . . . . . . . . . . . . . 16 (𝑈 ∩ Cat) ⊆ Cat
7371, 72syl6eqss 3815 . . . . . . . . . . . . . . 15 (𝜑𝐵 ⊆ Cat)
7473, 8sseldd 3762 . . . . . . . . . . . . . 14 (𝜑𝑋 ∈ Cat)
7574ad2antrr 717 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑋 ∈ Cat)
7661, 38, 75, 42idfu1 16805 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(idfunc𝑋))‘𝑥) = 𝑥)
7767, 70, 763eqtr3d 2807 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥)) = 𝑥)
7866fveq1d 6377 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑦) = ((1st ‘(idfunc𝑋))‘𝑦))
7938, 68, 69, 43cofu1 16809 . . . . . . . . . . . 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 16805 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(idfunc𝑋))‘𝑦) = 𝑦)
8178, 79, 803eqtr3d 2807 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑦)) = 𝑦)
8277, 81oveq12d 6860 . . . . . . . . . 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 6210 . . . . . . . . 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 223 . . . . . . . 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 6379 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (2nd ‘(idfunc𝑋)))
8685oveqd 6859 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))𝑦) = (𝑥(2nd ‘(idfunc𝑋))𝑦))
8738, 68, 69, 42, 43cofu2nd 16810 . . . . . . . . 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 16802 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd ‘(idfunc𝑋))𝑦) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
8986, 87, 883eqtr3d 2807 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
9023simprd 489 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
912, 25, 26, 27, 21, 14, 16, 15issect 16678 . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . 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 1174 . . . . . . . . . . . . 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 17016 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
95 eqid 2765 . . . . . . . . . . . . . . 15 (idfunc𝑌) = (idfunc𝑌)
965, 2, 27, 95, 4, 9catcid 17018 . . . . . . . . . . . . . 14 (𝜑 → ((Id‘𝐶)‘𝑌) = (idfunc𝑌))
9796adantr 472 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑌) = (idfunc𝑌))
9893, 94, 973eqtr3d 2807 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (idfunc𝑌))
9998adantr 472 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (idfunc𝑌))
10099fveq2d 6379 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (2nd ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (2nd ‘(idfunc𝑌)))
101100oveqd 6859 . . . . . . . . 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 16810 . . . . . . . . . 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 6860 . . . . . . . . . . 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 5452 . . . . . . . . . 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 2799 . . . . . . . . 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 717 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝐵 ⊆ Cat)
1079ad2antrr 717 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑌𝐵)
108106, 107sseldd 3762 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑌 ∈ Cat)
10995, 45, 108, 40, 55, 56idfu2nd 16802 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((1st𝐹)‘𝑥)(2nd ‘(idfunc𝑌))((1st𝐹)‘𝑦)) = ( I ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦))))
110101, 105, 1093eqtr3d 2807 . . . . . . . 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 6741 . . . . . . 7 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦)))
112111ralrimivva 3118 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ∀𝑥𝑅𝑦𝑅 (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦)))
11338, 39, 40isffth2 16841 . . . . . 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 578 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹))
115 df-br 4810 . . . . 5 ((1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹) ↔ ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
116114, 115sylib 209 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
11735, 116eqeltrd 2844 . . 3 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
11838, 45, 37funcf1 16791 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹):𝑅𝑆)
11945, 38, 52funcf1 16791 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)):𝑆𝑅)
12064fveq2d 6379 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (1st ‘(idfunc𝑋)))
12138, 33, 50cofu1st 16808 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∘ (1st𝐹)))
12274adantr 472 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ Cat)
12361, 38, 122idfu1st 16804 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(idfunc𝑋)) = ( I ↾ 𝑅))
124120, 121, 1233eqtr3d 2807 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∘ (1st𝐹)) = ( I ↾ 𝑅))
12598fveq2d 6379 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (1st ‘(idfunc𝑌)))
12645, 50, 33cofu1st 16808 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ((1st𝐹) ∘ (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
12773, 9sseldd 3762 . . . . . . 7 (𝜑𝑌 ∈ Cat)
128127adantr 472 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ Cat)
12995, 45, 128idfu1st 16804 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(idfunc𝑌)) = ( I ↾ 𝑆))
130125, 126, 1293eqtr3d 2807 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((1st𝐹) ∘ (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ( I ↾ 𝑆))
131118, 119, 124, 130fcof1od 6741 . . 3 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹):𝑅1-1-onto𝑆)
132117, 131jca 507 . 2 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆))
1337adantr 472 . . 3 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐶 ∈ Cat)
1348adantr 472 . . 3 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝑋𝐵)
1359adantr 472 . . 3 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝑌𝐵)
136 inss1 3992 . . . . . . 7 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Full 𝑌)
137 fullfunc 16831 . . . . . . 7 (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
138136, 137sstri 3770 . . . . . 6 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Func 𝑌)
139 simprl 787 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
140138, 139sseldi 3759 . . . . 5 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 ∈ (𝑋 Func 𝑌))
1411, 140, 34sylancr 581 . . . 4 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
1424adantr 472 . . . . 5 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝑈𝑉)
143 eqid 2765 . . . . 5 (𝑥𝑆, 𝑦𝑆(((1st𝐹)‘𝑥)(2nd𝐹)((1st𝐹)‘𝑦))) = (𝑥𝑆, 𝑦𝑆(((1st𝐹)‘𝑥)(2nd𝐹)((1st𝐹)‘𝑦)))
144141, 139eqeltrrd 2845 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
145144, 115sylibr 225 . . . . 5 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → (1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹))
146 simprr 789 . . . . 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 17021 . . . 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 4831 . . 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 16691 . 2 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 ∈ (𝑋𝐼𝑌))
150132, 149impbida 835 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  cin 3731  wss 3732  cop 4340   class class class wbr 4809   I cid 5184  ccnv 5276  dom cdm 5277  cres 5279  ccom 5281  Rel wrel 5282  Fun wfun 6062  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  cmpt2 6844  1st c1st 7364  2nd c2nd 7365  Basecbs 16130  Hom chom 16225  compcco 16226  Catccat 16590  Idccid 16591  Sectcsect 16669  Invcinv 16670  Isociso 16671   Func cfunc 16779  idfunccidfu 16780  func ccofu 16781   Full cful 16827   Faith cfth 16828  CatCatccatc 17009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-hom 16238  df-cco 16239  df-cat 16594  df-cid 16595  df-sect 16672  df-inv 16673  df-iso 16674  df-func 16783  df-idfu 16784  df-cofu 16785  df-full 16829  df-fth 16830  df-catc 17010
This theorem is referenced by:  yoniso  17191
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