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Theorem catciso 18061

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	⊢ (𝜑 → 𝑈 ∈ 𝑉)
catciso.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
catciso.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
catciso.i	⊢ 𝐼 = (Iso‘𝐶)

**Assertion**
Ref	Expression
catciso	⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)))

Proof of Theorem catciso Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 17812	. . . . 5 ⊢ Rel (𝑋 Func 𝑌)
2		catciso.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
4		catciso.u	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
5		catciso.c	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = (CatCat‘𝑈)
6	5	catccat 18058	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
7	4, 6	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ Cat)
8		catciso.x	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		catciso.y	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		catciso.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = (Iso‘𝐶)
11	2, 3, 7, 8, 9, 10	isoval 17712	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
12	11	eleq2d 2820	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
13	12	biimpa 478	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))
14	7	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ Cat)
15	8	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝐵)
16	9	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝐵)
17	2, 3, 14, 15, 16	invfun 17711	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → Fun (𝑋(Inv‘𝐶)𝑌))
18		funfvbrb 7053	. . . . . . . . . . . 12 ⊢ (Fun (𝑋(Inv‘𝐶)𝑌) → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
19	17, 18	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
20	13, 19	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
21		eqid 2733	. . . . . . . . . . 11 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
22	2, 3, 14, 15, 16, 21	isinv 17707	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (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)))
23	20, 22	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
24	23	simpld 496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
25		eqid 2733	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
26		eqid 2733	. . . . . . . . 9 ⊢ (comp‘𝐶) = (comp‘𝐶)
27		eqid 2733	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
28	2, 25, 26, 27, 21, 14, 15, 16	issect 17700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (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))))
29	24, 28	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (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)))
30	29	simp1d 1143	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
31	5, 2, 4, 25, 8, 9	catchom 18053	. . . . . . 7 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
32	31	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
33	30, 32	eleqtrd 2836	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ (𝑋 Func 𝑌))
34		1st2nd 8025	. . . . 5 ⊢ ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
35	1, 33, 34	sylancr 588	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
36		1st2ndbr 8028	. . . . . . 7 ⊢ ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → (1^st ‘𝐹)(𝑋 Func 𝑌)(2^nd ‘𝐹))
37	1, 33, 36	sylancr 588	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘𝐹)(𝑋 Func 𝑌)(2^nd ‘𝐹))
38		catciso.r	. . . . . . . . 9 ⊢ 𝑅 = (Base‘𝑋)
39		eqid 2733	. . . . . . . . 9 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
40		eqid 2733	. . . . . . . . 9 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
41	37	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (1^st ‘𝐹)(𝑋 Func 𝑌)(2^nd ‘𝐹))
42		simprl 770	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
43		simprr 772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
44	38, 39, 40, 41, 42, 43	funcf2 17818	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2^nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)⟶(((1^st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1^st ‘𝐹)‘𝑦)))
45		catciso.s	. . . . . . . . . 10 ⊢ 𝑆 = (Base‘𝑌)
46		relfunc 17812	. . . . . . . . . . . 12 ⊢ Rel (𝑌 Func 𝑋)
47	29	simp2d 1144	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
48	5, 2, 4, 25, 9, 8	catchom 18053	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
49	48	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
50	47, 49	eleqtrd 2836	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋))
51		1st2ndbr 8028	. . . . . . . . . . . 12 ⊢ ((Rel (𝑌 Func 𝑋) ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋)) → (1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
52	46, 50, 51	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
53	52	adantr 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
54	38, 45, 41	funcf1 17816	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (1^st ‘𝐹):𝑅⟶𝑆)
55	54, 42	ffvelcdmd 7088	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1^st ‘𝐹)‘𝑥) ∈ 𝑆)
56	54, 43	ffvelcdmd 7088	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1^st ‘𝐹)‘𝑦) ∈ 𝑆)
57	45, 40, 39, 53, 55, 56	funcf2 17818	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1^st ‘𝐹)‘𝑥)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1^st ‘𝐹)‘𝑦)):(((1^st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1^st ‘𝐹)‘𝑦))⟶(((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑥))(Hom ‘𝑋)((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑦))))
58	29	simp3d 1145	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
59	4	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝑈 ∈ 𝑉)
60	5, 2, 59, 26, 15, 16, 15, 33, 50	catcco 18055	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹))
61		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (id_func‘𝑋) = (id_func‘𝑋)
62	5, 2, 27, 61, 4, 8	catcid 18057	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) = (id_func‘𝑋))
63	62	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑋) = (id_func‘𝑋))
64	58, 60, 63	3eqtr3d 2781	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹) = (id_func‘𝑋))
65	64	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹) = (id_func‘𝑋))
66	65	fveq2d 6896	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (1^st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹)) = (1^st ‘(id_func‘𝑋)))
67	66	fveq1d 6894	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1^st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹))‘𝑥) = ((1^st ‘(id_func‘𝑋))‘𝑥))
68	33	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝐹 ∈ (𝑋 Func 𝑌))
69	50	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋))
70	38, 68, 69, 42	cofu1 17834	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1^st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹))‘𝑥) = ((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑥)))
71	5, 2, 4	catcbas 18051	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
72		inss2 4230	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∩ Cat) ⊆ Cat
73	71, 72	eqsstrdi 4037	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ⊆ Cat)
74	73, 8	sseldd 3984	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ Cat)
75	74	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑋 ∈ Cat)
76	61, 38, 75, 42	idfu1 17830	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1^st ‘(id_func‘𝑋))‘𝑥) = 𝑥)
77	67, 70, 76	3eqtr3d 2781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑥)) = 𝑥)
78	66	fveq1d 6894	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1^st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹))‘𝑦) = ((1^st ‘(id_func‘𝑋))‘𝑦))
79	38, 68, 69, 43	cofu1 17834	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1^st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹))‘𝑦) = ((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑦)))
80	61, 38, 75, 43	idfu1 17830	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1^st ‘(id_func‘𝑋))‘𝑦) = 𝑦)
81	78, 79, 80	3eqtr3d 2781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑦)) = 𝑦)
82	77, 81	oveq12d 7427	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑥))(Hom ‘𝑋)((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑦))) = (𝑥(Hom ‘𝑋)𝑦))
83	82	feq3d 6705	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((((1^st ‘𝐹)‘𝑥)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1^st ‘𝐹)‘𝑦)):(((1^st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1^st ‘𝐹)‘𝑦))⟶(((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑥))(Hom ‘𝑋)((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑦))) ↔ (((1^st ‘𝐹)‘𝑥)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1^st ‘𝐹)‘𝑦)):(((1^st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1^st ‘𝐹)‘𝑦))⟶(𝑥(Hom ‘𝑋)𝑦)))
84	57, 83	mpbid 231	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1^st ‘𝐹)‘𝑥)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1^st ‘𝐹)‘𝑦)):(((1^st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1^st ‘𝐹)‘𝑦))⟶(𝑥(Hom ‘𝑋)𝑦))
85	65	fveq2d 6896	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (2^nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹)) = (2^nd ‘(id_func‘𝑋)))
86	85	oveqd 7426	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2^nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹))𝑦) = (𝑥(2^nd ‘(id_func‘𝑋))𝑦))
87	38, 68, 69, 42, 43	cofu2nd 17835	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2^nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹))𝑦) = ((((1^st ‘𝐹)‘𝑥)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)))
88	61, 38, 75, 39, 42, 43	idfu2nd 17827	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2^nd ‘(id_func‘𝑋))𝑦) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
89	86, 87, 88	3eqtr3d 2781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((((1^st ‘𝐹)‘𝑥)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦)) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
90	23	simprd 497	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
91	2, 25, 26, 27, 21, 14, 16, 15	issect 17700	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 ∈ (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))))
92	90, 91	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∈ (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)))
93	92	simp3d 1145	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = ((Id‘𝐶)‘𝑌))
94	5, 2, 59, 26, 16, 15, 16, 50, 33	catcco 18055	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (𝐹 ∘_func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
95		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (id_func‘𝑌) = (id_func‘𝑌)
96	5, 2, 27, 95, 4, 9	catcid 18057	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) = (id_func‘𝑌))
97	96	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑌) = (id_func‘𝑌))
98	93, 94, 97	3eqtr3d 2781	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹 ∘_func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (id_func‘𝑌))
99	98	adantr 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝐹 ∘_func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (id_func‘𝑌))
100	99	fveq2d 6896	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (2^nd ‘(𝐹 ∘_func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (2^nd ‘(id_func‘𝑌)))
101	100	oveqd 7426	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1^st ‘𝐹)‘𝑥)(2^nd ‘(𝐹 ∘_func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))((1^st ‘𝐹)‘𝑦)) = (((1^st ‘𝐹)‘𝑥)(2^nd ‘(id_func‘𝑌))((1^st ‘𝐹)‘𝑦)))
102	45, 69, 68, 55, 56	cofu2nd 17835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1^st ‘𝐹)‘𝑥)(2^nd ‘(𝐹 ∘_func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))((1^st ‘𝐹)‘𝑦)) = ((((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑥))(2^nd ‘𝐹)((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑦))) ∘ (((1^st ‘𝐹)‘𝑥)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1^st ‘𝐹)‘𝑦))))
103	77, 81	oveq12d 7427	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑥))(2^nd ‘𝐹)((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑦))) = (𝑥(2^nd ‘𝐹)𝑦))
104	103	coeq1d 5862	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑥))(2^nd ‘𝐹)((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1^st ‘𝐹)‘𝑦))) ∘ (((1^st ‘𝐹)‘𝑥)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1^st ‘𝐹)‘𝑦))) = ((𝑥(2^nd ‘𝐹)𝑦) ∘ (((1^st ‘𝐹)‘𝑥)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1^st ‘𝐹)‘𝑦))))
105	102, 104	eqtrd 2773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1^st ‘𝐹)‘𝑥)(2^nd ‘(𝐹 ∘_func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))((1^st ‘𝐹)‘𝑦)) = ((𝑥(2^nd ‘𝐹)𝑦) ∘ (((1^st ‘𝐹)‘𝑥)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1^st ‘𝐹)‘𝑦))))
106	73	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝐵 ⊆ Cat)
107	9	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑌 ∈ 𝐵)
108	106, 107	sseldd 3984	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑌 ∈ Cat)
109	95, 45, 108, 40, 55, 56	idfu2nd 17827	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1^st ‘𝐹)‘𝑥)(2^nd ‘(id_func‘𝑌))((1^st ‘𝐹)‘𝑦)) = ( I ↾ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1^st ‘𝐹)‘𝑦))))
110	101, 105, 109	3eqtr3d 2781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥(2^nd ‘𝐹)𝑦) ∘ (((1^st ‘𝐹)‘𝑥)(2^nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1^st ‘𝐹)‘𝑦))) = ( I ↾ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1^st ‘𝐹)‘𝑦))))
111	44, 84, 89, 110	fcof1od 7292	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2^nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1^st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1^st ‘𝐹)‘𝑦)))
112	111	ralrimivva 3201	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝑥(2^nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1^st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1^st ‘𝐹)‘𝑦)))
113	38, 39, 40	isffth2 17867	. . . . . 6 ⊢ ((1^st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2^nd ‘𝐹) ↔ ((1^st ‘𝐹)(𝑋 Func 𝑌)(2^nd ‘𝐹) ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝑥(2^nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1^st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1^st ‘𝐹)‘𝑦))))
114	37, 112, 113	sylanbrc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2^nd ‘𝐹))
115		df-br 5150	. . . . 5 ⊢ ((1^st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2^nd ‘𝐹) ↔ ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
116	114, 115	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
117	35, 116	eqeltrd 2834	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
118	38, 45, 37	funcf1 17816	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘𝐹):𝑅⟶𝑆)
119	45, 38, 52	funcf1 17816	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)):𝑆⟶𝑅)
120	64	fveq2d 6896	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹)) = (1^st ‘(id_func‘𝑋)))
121	38, 33, 50	cofu1st 17833	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘_func 𝐹)) = ((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∘ (1^st ‘𝐹)))
122	74	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ Cat)
123	61, 38, 122	idfu1st 17829	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘(id_func‘𝑋)) = ( I ↾ 𝑅))
124	120, 121, 123	3eqtr3d 2781	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∘ (1^st ‘𝐹)) = ( I ↾ 𝑅))
125	98	fveq2d 6896	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘(𝐹 ∘_func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (1^st ‘(id_func‘𝑌)))
126	45, 50, 33	cofu1st 17833	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘(𝐹 ∘_func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ((1^st ‘𝐹) ∘ (1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
127	73, 9	sseldd 3984	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ Cat)
128	127	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ Cat)
129	95, 45, 128	idfu1st 17829	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘(id_func‘𝑌)) = ( I ↾ 𝑆))
130	125, 126, 129	3eqtr3d 2781	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((1^st ‘𝐹) ∘ (1^st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ( I ↾ 𝑆))
131	118, 119, 124, 130	fcof1od 7292	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1^st ‘𝐹):𝑅–1-1-onto→𝑆)
132	117, 131	jca 513	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆))
133	7	adantr 482	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝐶 ∈ Cat)
134	8	adantr 482	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝑋 ∈ 𝐵)
135	9	adantr 482	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝑌 ∈ 𝐵)
136		inss1 4229	. . . . . . 7 ⊢ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Full 𝑌)
137		fullfunc 17857	. . . . . . 7 ⊢ (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
138	136, 137	sstri 3992	. . . . . 6 ⊢ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Func 𝑌)
139		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
140	138, 139	sselid 3981	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝐹 ∈ (𝑋 Func 𝑌))
141	1, 140, 34	sylancr 588	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
142	4	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝑈 ∈ 𝑉)
143		eqid 2733	. . . . 5 ⊢ (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ^◡((^◡(1^st ‘𝐹)‘𝑥)(2^nd ‘𝐹)(^◡(1^st ‘𝐹)‘𝑦))) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ^◡((^◡(1^st ‘𝐹)‘𝑥)(2^nd ‘𝐹)(^◡(1^st ‘𝐹)‘𝑦)))
144	141, 139	eqeltrrd 2835	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
145	144, 115	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → (1^st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2^nd ‘𝐹))
146		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → (1^st ‘𝐹):𝑅–1-1-onto→𝑆)
147	5, 2, 38, 45, 142, 134, 135, 3, 143, 145, 146	catcisolem 18060	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝑋(Inv‘𝐶)𝑌)⟨^◡(1^st ‘𝐹), (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ^◡((^◡(1^st ‘𝐹)‘𝑥)(2^nd ‘𝐹)(^◡(1^st ‘𝐹)‘𝑦)))⟩)
148	141, 147	eqbrtrd 5171	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝐹(𝑋(Inv‘𝐶)𝑌)⟨^◡(1^st ‘𝐹), (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ^◡((^◡(1^st ‘𝐹)‘𝑥)(2^nd ‘𝐹)(^◡(1^st ‘𝐹)‘𝑦)))⟩)
149	2, 3, 133, 134, 135, 10, 148	inviso1 17713	. 2 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝐹 ∈ (𝑋𝐼𝑌))
150	132, 149	impbida 800	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝐹):𝑅–1-1-onto→𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∩ cin 3948 ⊆ wss 3949 ⟨cop 4635 class class class wbr 5149 I cid 5574 ^◡ccnv 5676 dom cdm 5677 ↾ cres 5679 ∘ ccom 5681 Rel wrel 5682 Fun wfun 6538 ⟶wf 6540 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 1^st c1st 7973 2^nd c2nd 7974 Basecbs 17144 Hom chom 17208 compcco 17209 Catccat 17608 Idccid 17609 Sectcsect 17691 Invcinv 17692 Isociso 17693 Func cfunc 17804 id_funccidfu 17805 ∘_func ccofu 17806 Full cful 17853 Faith cfth 17854 CatCatccatc 18048

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-slot 17115 df-ndx 17127 df-base 17145 df-hom 17221 df-cco 17222 df-cat 17612 df-cid 17613 df-sect 17694 df-inv 17695 df-iso 17696 df-func 17808 df-idfu 17809 df-cofu 17810 df-full 17855 df-fth 17856 df-catc 18049

This theorem is referenced by: yoniso 18238 thincciso 47717

W3C validator