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Theorem catciso 16738
 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 16503 . . . . 5 Rel (𝑋 Func 𝑌)
2 catciso.b . . . . . . . . . . . . . 14 𝐵 = (Base‘𝐶)
3 eqid 2620 . . . . . . . . . . . . . 14 (Inv‘𝐶) = (Inv‘𝐶)
4 catciso.u . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
5 catciso.c . . . . . . . . . . . . . . . 16 𝐶 = (CatCat‘𝑈)
65catccat 16735 . . . . . . . . . . . . . . 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 16406 . . . . . . . . . . . . 13 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
1211eleq2d 2685 . . . . . . . . . . . 12 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
1312biimpa 501 . . . . . . . . . . 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 16405 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → Fun (𝑋(Inv‘𝐶)𝑌))
18 funfvbrb 6316 . . . . . . . . . . . 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 222 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
21 eqid 2620 . . . . . . . . . . 11 (Sect‘𝐶) = (Sect‘𝐶)
222, 3, 14, 15, 16, 21isinv 16401 . . . . . . . . . 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 222 . . . . . . . . 9 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
2423simpld 475 . . . . . . . 8 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
25 eqid 2620 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
26 eqid 2620 . . . . . . . . 9 (comp‘𝐶) = (comp‘𝐶)
27 eqid 2620 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
282, 25, 26, 27, 21, 14, 15, 16issect 16394 . . . . . . . 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 222 . . . . . . 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 1071 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
315, 2, 4, 25, 8, 9catchom 16730 . . . . . . 7 (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
3231adantr 481 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
3330, 32eleqtrd 2701 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ (𝑋 Func 𝑌))
34 1st2nd 7199 . . . . 5 ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
351, 33, 34sylancr 694 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
36 1st2ndbr 7202 . . . . . . 7 ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → (1st𝐹)(𝑋 Func 𝑌)(2nd𝐹))
371, 33, 36sylancr 694 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹)(𝑋 Func 𝑌)(2nd𝐹))
38 catciso.r . . . . . . . . 9 𝑅 = (Base‘𝑋)
39 eqid 2620 . . . . . . . . 9 (Hom ‘𝑋) = (Hom ‘𝑋)
40 eqid 2620 . . . . . . . . 9 (Hom ‘𝑌) = (Hom ‘𝑌)
4137adantr 481 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st𝐹)(𝑋 Func 𝑌)(2nd𝐹))
42 simprl 793 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥𝑅)
43 simprr 795 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦𝑅)
4438, 39, 40, 41, 42, 43funcf2 16509 . . . . . . . 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 16503 . . . . . . . . . . . 12 Rel (𝑌 Func 𝑋)
4729simp2d 1072 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
485, 2, 4, 25, 9, 8catchom 16730 . . . . . . . . . . . . . 14 (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
4948adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
5047, 49eleqtrd 2701 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋))
51 1st2ndbr 7202 . . . . . . . . . . . 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 694 . . . . . . . . . . 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 16507 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st𝐹):𝑅𝑆)
5554, 42ffvelrnd 6346 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st𝐹)‘𝑥) ∈ 𝑆)
5654, 43ffvelrnd 6346 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st𝐹)‘𝑦) ∈ 𝑆)
5745, 40, 39, 53, 55, 56funcf2 16509 . . . . . . . . 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 1073 . . . . . . . . . . . . . . . 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 16732 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))
61 eqid 2620 . . . . . . . . . . . . . . . . . 18 (idfunc𝑋) = (idfunc𝑋)
625, 2, 27, 61, 4, 8catcid 16734 . . . . . . . . . . . . . . . . 17 (𝜑 → ((Id‘𝐶)‘𝑋) = (idfunc𝑋))
6362adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑋) = (idfunc𝑋))
6458, 60, 633eqtr3d 2662 . . . . . . . . . . . . . . 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 6182 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (1st ‘(idfunc𝑋)))
6766fveq1d 6180 . . . . . . . . . . . 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 16525 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑥) = ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥)))
715, 2, 4catcbas 16728 . . . . . . . . . . . . . . . 16 (𝜑𝐵 = (𝑈 ∩ Cat))
72 inss2 3826 . . . . . . . . . . . . . . . 16 (𝑈 ∩ Cat) ⊆ Cat
7371, 72syl6eqss 3647 . . . . . . . . . . . . . . 15 (𝜑𝐵 ⊆ Cat)
7473, 8sseldd 3596 . . . . . . . . . . . . . 14 (𝜑𝑋 ∈ Cat)
7574ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑋 ∈ Cat)
7661, 38, 75, 42idfu1 16521 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(idfunc𝑋))‘𝑥) = 𝑥)
7767, 70, 763eqtr3d 2662 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑥)) = 𝑥)
7866fveq1d 6180 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑦) = ((1st ‘(idfunc𝑋))‘𝑦))
7938, 68, 69, 43cofu1 16525 . . . . . . . . . . . 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 16521 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘(idfunc𝑋))‘𝑦) = 𝑦)
8178, 79, 803eqtr3d 2662 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st𝐹)‘𝑦)) = 𝑦)
8277, 81oveq12d 6653 . . . . . . . . . 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 6019 . . . . . . . . 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 222 . . . . . . . 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 6182 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (2nd ‘(idfunc𝑋)))
8685oveqd 6652 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))𝑦) = (𝑥(2nd ‘(idfunc𝑋))𝑦))
8738, 68, 69, 42, 43cofu2nd 16526 . . . . . . . . 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 16518 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd ‘(idfunc𝑋))𝑦) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
8986, 87, 883eqtr3d 2662 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((((1st𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
9023simprd 479 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
912, 25, 26, 27, 21, 14, 16, 15issect 16394 . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . 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 1073 . . . . . . . . . . . . 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 16732 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
95 eqid 2620 . . . . . . . . . . . . . . 15 (idfunc𝑌) = (idfunc𝑌)
965, 2, 27, 95, 4, 9catcid 16734 . . . . . . . . . . . . . 14 (𝜑 → ((Id‘𝐶)‘𝑌) = (idfunc𝑌))
9796adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑌) = (idfunc𝑌))
9893, 94, 973eqtr3d 2662 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (idfunc𝑌))
9998adantr 481 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (idfunc𝑌))
10099fveq2d 6182 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (2nd ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (2nd ‘(idfunc𝑌)))
101100oveqd 6652 . . . . . . . . 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 16526 . . . . . . . . . 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 6653 . . . . . . . . . . 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 5272 . . . . . . . . . 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 2654 . . . . . . . . 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 761 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝐵 ⊆ Cat)
1079ad2antrr 761 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑌𝐵)
108106, 107sseldd 3596 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑌 ∈ Cat)
10995, 45, 108, 40, 55, 56idfu2nd 16518 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (((1st𝐹)‘𝑥)(2nd ‘(idfunc𝑌))((1st𝐹)‘𝑦)) = ( I ↾ (((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦))))
110101, 105, 1093eqtr3d 2662 . . . . . . . 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 6534 . . . . . . 7 (((𝜑𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦)))
112111ralrimivva 2968 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ∀𝑥𝑅𝑦𝑅 (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st𝐹)‘𝑥)(Hom ‘𝑌)((1st𝐹)‘𝑦)))
11338, 39, 40isffth2 16557 . . . . . 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 697 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹))
115 df-br 4645 . . . . 5 ((1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹) ↔ ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
116114, 115sylib 208 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
11735, 116eqeltrd 2699 . . 3 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
11838, 45, 37funcf1 16507 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹):𝑅𝑆)
11945, 38, 52funcf1 16507 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)):𝑆𝑅)
12064fveq2d 6182 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (1st ‘(idfunc𝑋)))
12138, 33, 50cofu1st 16524 . . . . 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 16520 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(idfunc𝑋)) = ( I ↾ 𝑅))
124120, 121, 1233eqtr3d 2662 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∘ (1st𝐹)) = ( I ↾ 𝑅))
12598fveq2d 6182 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (1st ‘(idfunc𝑌)))
12645, 50, 33cofu1st 16524 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(𝐹func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ((1st𝐹) ∘ (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
12773, 9sseldd 3596 . . . . . . 7 (𝜑𝑌 ∈ Cat)
128127adantr 481 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ Cat)
12995, 45, 128idfu1st 16520 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(idfunc𝑌)) = ( I ↾ 𝑆))
130125, 126, 1293eqtr3d 2662 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → ((1st𝐹) ∘ (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ( I ↾ 𝑆))
131118, 119, 124, 130fcof1od 6534 . . 3 ((𝜑𝐹 ∈ (𝑋𝐼𝑌)) → (1st𝐹):𝑅1-1-onto𝑆)
132117, 131jca 554 . 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 3825 . . . . . . 7 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Full 𝑌)
137 fullfunc 16547 . . . . . . 7 (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
138136, 137sstri 3604 . . . . . 6 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Func 𝑌)
139 simprl 793 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
140138, 139sseldi 3593 . . . . 5 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 ∈ (𝑋 Func 𝑌))
1411, 140, 34sylancr 694 . . . 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 2620 . . . . 5 (𝑥𝑆, 𝑦𝑆(((1st𝐹)‘𝑥)(2nd𝐹)((1st𝐹)‘𝑦))) = (𝑥𝑆, 𝑦𝑆(((1st𝐹)‘𝑥)(2nd𝐹)((1st𝐹)‘𝑦)))
144141, 139eqeltrrd 2700 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → ⟨(1st𝐹), (2nd𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
145144, 115sylibr 224 . . . . 5 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → (1st𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd𝐹))
146 simprr 795 . . . . 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 16737 . . . 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 4666 . . 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 16407 . 2 ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)) → 𝐹 ∈ (𝑋𝐼𝑌))
150132, 149impbida 876 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 196   ∧ wa 384   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988  ∀wral 2909   ∩ cin 3566   ⊆ wss 3567  ⟨cop 4174   class class class wbr 4644   I cid 5013  ◡ccnv 5103  dom cdm 5104   ↾ cres 5106   ∘ ccom 5108  Rel wrel 5109  Fun wfun 5870  ⟶wf 5872  –1-1-onto→wf1o 5875  ‘cfv 5876  (class class class)co 6635   ↦ cmpt2 6637  1st c1st 7151  2nd c2nd 7152  Basecbs 15838  Hom chom 15933  compcco 15934  Catccat 16306  Idccid 16307  Sectcsect 16385  Invcinv 16386  Isociso 16387   Func cfunc 16495  idfunccidfu 16496   ∘func ccofu 16497   Full cful 16543   Faith cfth 16544  CatCatccatc 16725 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-hom 15947  df-cco 15948  df-cat 16310  df-cid 16311  df-sect 16388  df-inv 16389  df-iso 16390  df-func 16499  df-idfu 16500  df-cofu 16501  df-full 16545  df-fth 16546  df-catc 16726 This theorem is referenced by:  yoniso  16906
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