Theorem catciso 18040

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. Note that "catciso.u" is redundant thanks to elbasfv 17147. (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)) ∧ (1st ‘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 17791	. . . . 5 ⊢ Rel (𝑋 Func 𝑌)
2		catciso.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
4		catciso.u	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
5		catciso.c	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = (CatCat‘𝑈)
6	5	catccat 18037	. . . . . . . . . . . . . . 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 17694	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
12	11	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
13	12	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))
14	7	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ Cat)
15	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝐵)
16	9	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝐵)
17	2, 3, 14, 15, 16	invfun 17693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → Fun (𝑋(Inv‘𝐶)𝑌))
18		funfvbrb 6998	. . . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
21		eqid 2737	. . . . . . . . . . 11 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
22	2, 3, 14, 15, 16, 21	isinv 17689	. . . . . . . . . 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 232	. . . . . . . . 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 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))
25		eqid 2737	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
26		eqid 2737	. . . . . . . . 9 ⊢ (comp‘𝐶) = (comp‘𝐶)
27		eqid 2737	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
28	2, 25, 26, 27, 21, 14, 15, 16	issect 17682	. . . . . . . 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 232	. . . . . . 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 18032	. . . . . . 7 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
32	31	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
33	30, 32	eleqtrd 2839	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ (𝑋 Func 𝑌))
34		1st2nd 7986	. . . . 5 ⊢ ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
35	1, 33, 34	sylancr 588	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
36		1st2ndbr 7989	. . . . . . 7 ⊢ ((Rel (𝑋 Func 𝑌) ∧ 𝐹 ∈ (𝑋 Func 𝑌)) → (1st ‘𝐹)(𝑋 Func 𝑌)(2nd ‘𝐹))
37	1, 33, 36	sylancr 588	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘𝐹)(𝑋 Func 𝑌)(2nd ‘𝐹))
38		catciso.r	. . . . . . . . 9 ⊢ 𝑅 = (Base‘𝑋)
39		eqid 2737	. . . . . . . . 9 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
40		eqid 2737	. . . . . . . . 9 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
41	37	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (1st ‘𝐹)(𝑋 Func 𝑌)(2nd ‘𝐹))
42		simprl 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
43		simprr 773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
44	38, 39, 40, 41, 42, 43	funcf2 17797	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
45		catciso.s	. . . . . . . . . 10 ⊢ 𝑆 = (Base‘𝑌)
46		relfunc 17791	. . . . . . . . . . . 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 18032	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
49	48	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
50	47, 49	eleqtrd 2839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋))
51		1st2ndbr 7989	. . . . . . . . . . . 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)))
52	46, 50, 51	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
53	52	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))(𝑌 Func 𝑋)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
54	38, 45, 41	funcf1 17795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (1st ‘𝐹):𝑅⟶𝑆)
55	54, 42	ffvelcdmd 7032	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1st ‘𝐹)‘𝑥) ∈ 𝑆)
56	54, 43	ffvelcdmd 7032	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1st ‘𝐹)‘𝑦) ∈ 𝑆)
57	45, 40, 39, 53, 55, 56	funcf2 17797	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1st ‘𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st ‘𝐹)‘𝑦)):(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦))⟶(((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑥))(Hom ‘𝑋)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘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 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝑈 ∈ 𝑉)
60	5, 2, 59, 26, 15, 16, 15, 33, 50	catcco 18034	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))
61		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (idfunc‘𝑋) = (idfunc‘𝑋)
62	5, 2, 27, 61, 4, 8	catcid 18036	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) = (idfunc‘𝑋))
63	62	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑋) = (idfunc‘𝑋))
64	58, 60, 63	3eqtr3d 2780	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹) = (idfunc‘𝑋))
65	64	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹) = (idfunc‘𝑋))
66	65	fveq2d 6839	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (1st ‘(idfunc‘𝑋)))
67	66	fveq1d 6837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑥) = ((1st ‘(idfunc‘𝑋))‘𝑥))
68	33	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝐹 ∈ (𝑋 Func 𝑌))
69	50	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∈ (𝑌 Func 𝑋))
70	38, 68, 69, 42	cofu1 17813	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑥) = ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑥)))
71	5, 2, 4	catcbas 18030	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
72		inss2 4191	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∩ Cat) ⊆ Cat
73	71, 72	eqsstrdi 3979	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ⊆ Cat)
74	73, 8	sseldd 3935	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ Cat)
75	74	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑋 ∈ Cat)
76	61, 38, 75, 42	idfu1 17809	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1st ‘(idfunc‘𝑋))‘𝑥) = 𝑥)
77	67, 70, 76	3eqtr3d 2780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑥)) = 𝑥)
78	66	fveq1d 6837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑦) = ((1st ‘(idfunc‘𝑋))‘𝑦))
79	38, 68, 69, 43	cofu1 17813	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))‘𝑦) = ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑦)))
80	61, 38, 75, 43	idfu1 17809	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1st ‘(idfunc‘𝑋))‘𝑦) = 𝑦)
81	78, 79, 80	3eqtr3d 2780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑦)) = 𝑦)
82	77, 81	oveq12d 7379	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑥))(Hom ‘𝑋)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑦))) = (𝑥(Hom ‘𝑋)𝑦))
83	82	feq3d 6648	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st ‘𝐹)‘𝑦)):(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦))⟶(((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑥))(Hom ‘𝑋)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑦))) ↔ (((1st ‘𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st ‘𝐹)‘𝑦)):(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦))⟶(𝑥(Hom ‘𝑋)𝑦)))
84	57, 83	mpbid 232	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1st ‘𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st ‘𝐹)‘𝑦)):(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦))⟶(𝑥(Hom ‘𝑋)𝑦))
85	65	fveq2d 6839	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (2nd ‘(idfunc‘𝑋)))
86	85	oveqd 7378	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))𝑦) = (𝑥(2nd ‘(idfunc‘𝑋))𝑦))
87	38, 68, 69, 42, 43	cofu2nd 17814	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2nd ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹))𝑦) = ((((1st ‘𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))
88	61, 38, 75, 39, 42, 43	idfu2nd 17806	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2nd ‘(idfunc‘𝑋))𝑦) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
89	86, 87, 88	3eqtr3d 2780	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)) = ( I ↾ (𝑥(Hom ‘𝑋)𝑦)))
90	23	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((𝑋(Inv‘𝐶)𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
91	2, 25, 26, 27, 21, 14, 16, 15	issect 17682	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 18034	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (𝐹 ∘func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))
95		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (idfunc‘𝑌) = (idfunc‘𝑌)
96	5, 2, 27, 95, 4, 9	catcid 18036	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) = (idfunc‘𝑌))
97	96	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((Id‘𝐶)‘𝑌) = (idfunc‘𝑌))
98	93, 94, 97	3eqtr3d 2780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹 ∘func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (idfunc‘𝑌))
99	98	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝐹 ∘func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)) = (idfunc‘𝑌))
100	99	fveq2d 6839	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (2nd ‘(𝐹 ∘func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (2nd ‘(idfunc‘𝑌)))
101	100	oveqd 7378	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1st ‘𝐹)‘𝑥)(2nd ‘(𝐹 ∘func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))((1st ‘𝐹)‘𝑦)) = (((1st ‘𝐹)‘𝑥)(2nd ‘(idfunc‘𝑌))((1st ‘𝐹)‘𝑦)))
102	45, 69, 68, 55, 56	cofu2nd 17814	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1st ‘𝐹)‘𝑥)(2nd ‘(𝐹 ∘func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))((1st ‘𝐹)‘𝑦)) = ((((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑥))(2nd ‘𝐹)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑦))) ∘ (((1st ‘𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st ‘𝐹)‘𝑦))))
103	77, 81	oveq12d 7379	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑥))(2nd ‘𝐹)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑦))) = (𝑥(2nd ‘𝐹)𝑦))
104	103	coeq1d 5811	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑥))(2nd ‘𝐹)((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))‘((1st ‘𝐹)‘𝑦))) ∘ (((1st ‘𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st ‘𝐹)‘𝑦))) = ((𝑥(2nd ‘𝐹)𝑦) ∘ (((1st ‘𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st ‘𝐹)‘𝑦))))
105	102, 104	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1st ‘𝐹)‘𝑥)(2nd ‘(𝐹 ∘func ((𝑋(Inv‘𝐶)𝑌)‘𝐹)))((1st ‘𝐹)‘𝑦)) = ((𝑥(2nd ‘𝐹)𝑦) ∘ (((1st ‘𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st ‘𝐹)‘𝑦))))
106	73	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝐵 ⊆ Cat)
107	9	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑌 ∈ 𝐵)
108	106, 107	sseldd 3935	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑌 ∈ Cat)
109	95, 45, 108, 40, 55, 56	idfu2nd 17806	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (((1st ‘𝐹)‘𝑥)(2nd ‘(idfunc‘𝑌))((1st ‘𝐹)‘𝑦)) = ( I ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦))))
110	101, 105, 109	3eqtr3d 2780	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥(2nd ‘𝐹)𝑦) ∘ (((1st ‘𝐹)‘𝑥)(2nd ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))((1st ‘𝐹)‘𝑦))) = ( I ↾ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦))))
111	44, 84, 89, 110	fcof1od 7243	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
112	111	ralrimivva 3180	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
113	38, 39, 40	isffth2 17847	. . . . . 6 ⊢ ((1st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd ‘𝐹) ↔ ((1st ‘𝐹)(𝑋 Func 𝑌)(2nd ‘𝐹) ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦))))
114	37, 112, 113	sylanbrc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd ‘𝐹))
115		df-br 5100	. . . . 5 ⊢ ((1st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd ‘𝐹) ↔ ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
116	114, 115	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
117	35, 116	eqeltrd 2837	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
118	38, 45, 37	funcf1 17795	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘𝐹):𝑅⟶𝑆)
119	45, 38, 52	funcf1 17795	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)):𝑆⟶𝑅)
120	64	fveq2d 6839	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = (1st ‘(idfunc‘𝑋)))
121	38, 33, 50	cofu1st 17812	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(((𝑋(Inv‘𝐶)𝑌)‘𝐹) ∘func 𝐹)) = ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∘ (1st ‘𝐹)))
122	74	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ Cat)
123	61, 38, 122	idfu1st 17808	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(idfunc‘𝑋)) = ( I ↾ 𝑅))
124	120, 121, 123	3eqtr3d 2780	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹)) ∘ (1st ‘𝐹)) = ( I ↾ 𝑅))
125	98	fveq2d 6839	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(𝐹 ∘func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = (1st ‘(idfunc‘𝑌)))
126	45, 50, 33	cofu1st 17812	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(𝐹 ∘func ((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ((1st ‘𝐹) ∘ (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))))
127	73, 9	sseldd 3935	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ Cat)
128	127	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ Cat)
129	95, 45, 128	idfu1st 17808	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘(idfunc‘𝑌)) = ( I ↾ 𝑆))
130	125, 126, 129	3eqtr3d 2780	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → ((1st ‘𝐹) ∘ (1st ‘((𝑋(Inv‘𝐶)𝑌)‘𝐹))) = ( I ↾ 𝑆))
131	118, 119, 124, 130	fcof1od 7243	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (1st ‘𝐹):𝑅–1-1-onto→𝑆)
132	117, 131	jca 511	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐼𝑌)) → (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆))
133	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝐶 ∈ Cat)
134	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝑋 ∈ 𝐵)
135	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝑌 ∈ 𝐵)
136		inss1 4190	. . . . . . 7 ⊢ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Full 𝑌)
137		fullfunc 17837	. . . . . . 7 ⊢ (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
138	136, 137	sstri 3944	. . . . . 6 ⊢ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Func 𝑌)
139		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
140	138, 139	sselid 3932	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘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)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
142	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝑈 ∈ 𝑉)
143		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ◡((◡(1st ‘𝐹)‘𝑥)(2nd ‘𝐹)(◡(1st ‘𝐹)‘𝑦))) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ◡((◡(1st ‘𝐹)‘𝑥)(2nd ‘𝐹)(◡(1st ‘𝐹)‘𝑦)))
144	141, 139	eqeltrrd 2838	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
145	144, 115	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → (1st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd ‘𝐹))
146		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → (1st ‘𝐹):𝑅–1-1-onto→𝑆)
147	5, 2, 38, 45, 142, 134, 135, 3, 143, 145, 146	catcisolem 18039	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝑋(Inv‘𝐶)𝑌)⟨◡(1st ‘𝐹), (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ◡((◡(1st ‘𝐹)‘𝑥)(2nd ‘𝐹)(◡(1st ‘𝐹)‘𝑦)))⟩)
148	141, 147	eqbrtrd 5121	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝐹(𝑋(Inv‘𝐶)𝑌)⟨◡(1st ‘𝐹), (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ◡((◡(1st ‘𝐹)‘𝑥)(2nd ‘𝐹)(◡(1st ‘𝐹)‘𝑦)))⟩)
149	2, 3, 133, 134, 135, 10, 148	inviso1 17695	. 2 ⊢ ((𝜑 ∧ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)) → 𝐹 ∈ (𝑋𝐼𝑌))
150	132, 149	impbida 801	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3901   ⊆ wss 3902  ⟨cop 4587   class class class wbr 5099   I cid 5519  ^◡ccnv 5624  dom cdm 5625   ↾ cres 5627   ∘ ccom 5629  Rel wrel 5630  Fun wfun 6487  ⟶wf 6489  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363  1^st c1st 7934  2^nd c2nd 7935  Basecbs 17141  Hom chom 17193  compcco 17194  Catccat 17592  Idccid 17593  Sectcsect 17673  Invcinv 17674  Isociso 17675   Func cfunc 17783  id_funccidfu 17784   ∘_func ccofu 17785   Full cful 17833   Faith cfth 17834  CatCatccatc 18027

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683  ax-cnex 11087  ax-resscn 11088  ax-1cn 11089  ax-icn 11090  ax-addcl 11091  ax-addrcl 11092  ax-mulcl 11093  ax-mulrcl 11094  ax-mulcom 11095  ax-addass 11096  ax-mulass 11097  ax-distr 11098  ax-i2m1 11099  ax-1ne0 11100  ax-1rid 11101  ax-rnegex 11102  ax-rrecex 11103  ax-cnre 11104  ax-pre-lttri 11105  ax-pre-lttrn 11106  ax-pre-ltadd 11107  ax-pre-mulgt0 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-map 8770  df-ixp 8841  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12151  df-2 12213  df-3 12214  df-4 12215  df-5 12216  df-6 12217  df-7 12218  df-8 12219  df-9 12220  df-n0 12407  df-z 12494  df-dec 12613  df-uz 12757  df-fz 13429  df-struct 17079  df-slot 17114  df-ndx 17126  df-base 17142  df-hom 17206  df-cco 17207  df-cat 17596  df-cid 17597  df-sect 17676  df-inv 17677  df-iso 17678  df-func 17787  df-idfu 17788  df-cofu 17789  df-full 17835  df-fth 17836  df-catc 18028

This theorem is referenced by:  yoniso  18213  swapfiso  49607  catcisoi  49722  fucoppc  49732  thincciso  49775  thincciso2  49777  termcterm2  49836  diagciso  49861