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Theorem invfuc 18040

Description: If 𝑉(𝑥) is an inverse to 𝑈(𝑥) for each 𝑥, and 𝑈 is a natural transformation, then 𝑉 is also a natural transformation, and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)

**Hypotheses**
Ref	Expression
fuciso.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fuciso.b	⊢ 𝐵 = (Base‘𝐶)
fuciso.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fuciso.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
fuciso.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
fucinv.i	⊢ 𝐼 = (Inv‘𝑄)
fucinv.j	⊢ 𝐽 = (Inv‘𝐷)
invfuc.u	⊢ (𝜑 → 𝑈 ∈ (𝐹𝑁𝐺))
invfuc.v	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))𝑋)

**Assertion**
Ref	Expression
invfuc	⊢ (𝜑 → 𝑈(𝐹𝐼𝐺)(𝑥 ∈ 𝐵 ↦ 𝑋))

Distinct variable groups: 𝑥,𝐵 𝑥,𝐶 𝑥,𝐷 𝑥,𝐼 𝑥,𝐹 𝑥,𝐺 𝑥,𝐽 𝑥,𝑁 𝜑,𝑥 𝑥,𝑄 𝑥,𝑈 Allowed substitution hint: 𝑋(𝑥)

Proof of Theorem invfuc Dummy variables 𝑦 𝑓 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		invfuc.u	. 2 ⊢ (𝜑 → 𝑈 ∈ (𝐹𝑁𝐺))
2		invfuc.v	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))𝑋)
3		eqid 2762	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		fucinv.j	. . . . . . . . . 10 ⊢ 𝐽 = (Inv‘𝐷)
5		fuciso.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6		funcrcl 17926	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
7	5, 6	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8	7	simprd 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ Cat)
9	8	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
10		fuciso.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐶)
11		relfunc 17925	. . . . . . . . . . . . 13 ⊢ Rel (𝐶 Func 𝐷)
12		1st2ndbr 8037	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
13	11, 5, 12	sylancr 598	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
14	10, 3, 13	funcf1 17929	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
15	14	ffvelcdmda 7079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
16		fuciso.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
17		1st2ndbr 8037	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
18	11, 16, 17	sylancr 598	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
19	10, 3, 18	funcf1 17929	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
20	19	ffvelcdmda 7079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
21		eqid 2762	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22	3, 4, 9, 15, 20, 21	invss 17824	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ⊆ ((((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)) × (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥))))
23	22	ssbrd 5153	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))𝑋 → (𝑈‘𝑥)((((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)) × (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))𝑋))
24	2, 23	mpd 16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈‘𝑥)((((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)) × (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))𝑋)
25		brxp 5709	. . . . . . . 8 ⊢ ((𝑈‘𝑥)((((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)) × (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))𝑋 ↔ ((𝑈‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)) ∧ 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥))))
26	25	simprbi 502	. . . . . . 7 ⊢ ((𝑈‘𝑥)((((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)) × (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))𝑋 → 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
27	24, 26	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
28	27	ralrimiva 3156	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
29	10	fvexi 6895	. . . . . 6 ⊢ 𝐵 ∈ V
30		mptelixpg 8931	. . . . . 6 ⊢ (𝐵 ∈ V → ((𝑥 ∈ 𝐵 ↦ 𝑋) ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)) ↔ ∀𝑥 ∈ 𝐵 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥))))
31	29, 30	ax-mp 5	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝑋) ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)) ↔ ∀𝑥 ∈ 𝐵 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
32	28, 31	sylibr 237	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
33		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑦))
34		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑦))
35	33, 34	oveq12d 7430	. . . . 5 ⊢ (𝑥 = 𝑦 → (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)) = (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)))
36	35	cbvixpv 8911	. . . 4 ⊢ X𝑥 ∈ 𝐵 (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)) = X𝑦 ∈ 𝐵 (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦))
37	32, 36	eleqtrdi 2872	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ X𝑦 ∈ 𝐵 (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)))
38		simpr2 1213	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ 𝐵)
39		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
40		eqid 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑋) = (𝑥 ∈ 𝐵 ↦ 𝑋)
41	40	fvmpt2 7001	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥))) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
42	39, 27, 41	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
43	2, 42	breqtrrd 5138	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥))
44	43	ralrimiva 3156	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥))
45	44	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥))
46		nfcv 2924	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑈‘𝑧)
47		nfcv 2924	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))
48		nffvmpt1 6892	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)
49	46, 47, 48	nfbr 5157	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)
50		fveq2 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑈‘𝑥) = (𝑈‘𝑧))
51		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑧))
52		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑧))
53	51, 52	oveq12d 7430	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) = (((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧)))
54		fveq2 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧))
55	50, 53, 54	breq123d 5122	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) ↔ (𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)))
56	49, 55	rspc 3568	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) → (𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)))
57	38, 45, 56	sylc 66	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧))
58	8	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
59	14	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
60	59, 38	ffvelcdmd 7080	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐹)‘𝑧) ∈ (Base‘𝐷))
61	19	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
62	61, 38	ffvelcdmd 7080	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐺)‘𝑧) ∈ (Base‘𝐷))
63		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
64	3, 4, 58, 60, 62, 63	isinv 17823	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧) ↔ ((𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)(Sect‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧) ∧ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(((1^st ‘𝐺)‘𝑧)(Sect‘𝐷)((1^st ‘𝐹)‘𝑧))(𝑈‘𝑧))))
65	57, 64	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)(Sect‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧) ∧ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(((1^st ‘𝐺)‘𝑧)(Sect‘𝐷)((1^st ‘𝐹)‘𝑧))(𝑈‘𝑧)))
66	65	simpld 499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)(Sect‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧))
67		eqid 2762	. . . . . . . . . . . 12 ⊢ (comp‘𝐷) = (comp‘𝐷)
68		eqid 2762	. . . . . . . . . . . 12 ⊢ (Id‘𝐷) = (Id‘𝐷)
69	3, 21, 67, 68, 63, 58, 60, 62	issect 17816	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)(Sect‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧) ↔ ((𝑈‘𝑧) ∈ (((1^st ‘𝐹)‘𝑧)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑧)) ∧ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧) ∈ (((1^st ‘𝐺)‘𝑧)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑧)) ∧ (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑧), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))(𝑈‘𝑧)) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑧)))))
70	66, 69	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈‘𝑧) ∈ (((1^st ‘𝐹)‘𝑧)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑧)) ∧ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧) ∈ (((1^st ‘𝐺)‘𝑧)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑧)) ∧ (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑧), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))(𝑈‘𝑧)) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑧))))
71	70	simp3d 1161	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑧), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))(𝑈‘𝑧)) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑧)))
72	71	oveq1d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑧), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))(𝑈‘𝑧))(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑦(2^nd ‘𝐹)𝑧)‘𝑓)) = (((Id‘𝐷)‘((1^st ‘𝐹)‘𝑧))(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑦(2^nd ‘𝐹)𝑧)‘𝑓)))
73		simpr1 1212	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ 𝐵)
74	59, 73	ffvelcdmd 7080	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
75		eqid 2762	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
76	13	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
77	10, 75, 21, 76, 73, 38	funcf2 17931	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2^nd ‘𝐹)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1^st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑧)))
78		simpr3 1214	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
79	77, 78	ffvelcdmd 7080	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2^nd ‘𝐹)𝑧)‘𝑓) ∈ (((1^st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑧)))
80	3, 21, 68, 58, 74, 67, 60, 79	catlid 17745	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((Id‘𝐷)‘((1^st ‘𝐹)‘𝑧))(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑦(2^nd ‘𝐹)𝑧)‘𝑓)) = ((𝑦(2^nd ‘𝐹)𝑧)‘𝑓))
81	72, 80	eqtr2d 2798	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2^nd ‘𝐹)𝑧)‘𝑓) = ((((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑧), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))(𝑈‘𝑧))(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑦(2^nd ‘𝐹)𝑧)‘𝑓)))
82		fuciso.n	. . . . . . . . 9 ⊢ 𝑁 = (𝐶 Nat 𝐷)
83	1	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (𝐹𝑁𝐺))
84	82, 83	nat1st2nd 18017	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
85	82, 84, 10, 21, 38	natcl 18019	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈‘𝑧) ∈ (((1^st ‘𝐹)‘𝑧)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑧)))
86	70	simp2d 1160	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧) ∈ (((1^st ‘𝐺)‘𝑧)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑧)))
87	3, 21, 67, 58, 74, 60, 62, 79, 85, 60, 86	catass 17748	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑧), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))(𝑈‘𝑧))(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑦(2^nd ‘𝐹)𝑧)‘𝑓)) = (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑈‘𝑧)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑦(2^nd ‘𝐹)𝑧)‘𝑓))))
88	82, 84, 10, 75, 67, 73, 38, 78	nati 18021	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈‘𝑧)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑦(2^nd ‘𝐹)𝑧)‘𝑓)) = (((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))(𝑈‘𝑦)))
89	88	oveq2d 7428	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑈‘𝑧)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑦(2^nd ‘𝐹)𝑧)‘𝑓))) = (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))(((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))(𝑈‘𝑦))))
90	81, 87, 89	3eqtrd 2801	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2^nd ‘𝐹)𝑧)‘𝑓) = (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))(((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))(𝑈‘𝑦))))
91	90	oveq1d 7427	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2^nd ‘𝐹)𝑧)‘𝑓)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)) = ((((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))(((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))(𝑈‘𝑦)))(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)))
92	61, 73	ffvelcdmd 7080	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
93		nfcv 2924	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑈‘𝑦)
94		nfcv 2924	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))
95		nffvmpt1 6892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)
96	93, 94, 95	nfbr 5157	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)
97		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑈‘𝑥) = (𝑈‘𝑦))
98	34, 33	oveq12d 7430	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) = (((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦)))
99		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))
100	97, 98, 99	breq123d 5122	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) ↔ (𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)))
101	96, 100	rspc 3568	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) → (𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)))
102	73, 45, 101	sylc 66	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))
103	3, 4, 58, 74, 92, 63	isinv 17823	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦) ↔ ((𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)(Sect‘𝐷)((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦) ∧ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)(((1^st ‘𝐺)‘𝑦)(Sect‘𝐷)((1^st ‘𝐹)‘𝑦))(𝑈‘𝑦))))
104	102, 103	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)(Sect‘𝐷)((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦) ∧ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)(((1^st ‘𝐺)‘𝑦)(Sect‘𝐷)((1^st ‘𝐹)‘𝑦))(𝑈‘𝑦)))
105	104	simprd 500	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)(((1^st ‘𝐺)‘𝑦)(Sect‘𝐷)((1^st ‘𝐹)‘𝑦))(𝑈‘𝑦))
106	3, 21, 67, 68, 63, 58, 92, 74	issect 17816	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)(((1^st ‘𝐺)‘𝑦)(Sect‘𝐷)((1^st ‘𝐹)‘𝑦))(𝑈‘𝑦) ↔ (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦) ∈ (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)) ∧ (𝑈‘𝑦) ∈ (((1^st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑦)) ∧ ((𝑈‘𝑦)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)) = ((Id‘𝐷)‘((1^st ‘𝐺)‘𝑦)))))
107	105, 106	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦) ∈ (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)) ∧ (𝑈‘𝑦) ∈ (((1^st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑦)) ∧ ((𝑈‘𝑦)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)) = ((Id‘𝐷)‘((1^st ‘𝐺)‘𝑦))))
108	107	simp1d 1159	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦) ∈ (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)))
109	107	simp2d 1160	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈‘𝑦) ∈ (((1^st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑦)))
110	18	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
111	10, 75, 21, 110, 73, 38	funcf2 17931	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2^nd ‘𝐺)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑧)))
112	111, 78	ffvelcdmd 7080	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2^nd ‘𝐺)𝑧)‘𝑓) ∈ (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑧)))
113	3, 21, 67, 58, 74, 92, 62, 109, 112	catcocl 17747	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))(𝑈‘𝑦)) ∈ (((1^st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑧)))
114	3, 21, 67, 58, 92, 74, 62, 108, 113, 60, 86	catass 17748	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))(((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))(𝑈‘𝑦)))(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)) = (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))(𝑈‘𝑦))(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))))
115	82, 84, 10, 21, 73	natcl 18019	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈‘𝑦) ∈ (((1^st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑦)))
116	3, 21, 67, 58, 92, 74, 92, 108, 115, 62, 112	catass 17748	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))(𝑈‘𝑦))(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)) = (((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑈‘𝑦)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))))
117	107	simp3d 1161	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈‘𝑦)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)) = ((Id‘𝐷)‘((1^st ‘𝐺)‘𝑦)))
118	117	oveq2d 7428	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑈‘𝑦)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))) = (((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))((Id‘𝐷)‘((1^st ‘𝐺)‘𝑦))))
119	3, 21, 68, 58, 92, 67, 62, 112	catrid 17746	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))((Id‘𝐷)‘((1^st ‘𝐺)‘𝑦))) = ((𝑦(2^nd ‘𝐺)𝑧)‘𝑓))
120	116, 118, 119	3eqtrd 2801	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))(𝑈‘𝑦))(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)) = ((𝑦(2^nd ‘𝐺)𝑧)‘𝑓))
121	120	oveq2d 7428	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)(⟨((1^st ‘𝐹)‘𝑦), ((1^st ‘𝐺)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))(𝑈‘𝑦))(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))) = (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)))
122	91, 114, 121	3eqtrrd 2802	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)) = (((𝑦(2^nd ‘𝐹)𝑧)‘𝑓)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)))
123	122	ralrimivvva 3210	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧)(((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)) = (((𝑦(2^nd ‘𝐹)𝑧)‘𝑓)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)))
124	82, 10, 75, 21, 67, 16, 5	isnat2 18014	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝑋) ∈ (𝐺𝑁𝐹) ↔ ((𝑥 ∈ 𝐵 ↦ 𝑋) ∈ X𝑦 ∈ 𝐵 (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧)(((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐺)‘𝑧)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑦(2^nd ‘𝐺)𝑧)‘𝑓)) = (((𝑦(2^nd ‘𝐹)𝑧)‘𝑓)(⟨((1^st ‘𝐺)‘𝑦), ((1^st ‘𝐹)‘𝑦)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)))))
125	37, 123, 124	mpbir2and 725	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ (𝐺𝑁𝐹))
126		nfv 1943	. . . 4 ⊢ Ⅎ𝑦(𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥)
127	126, 96, 100	cbvralw 3306	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) ↔ ∀𝑦 ∈ 𝐵 (𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))
128	44, 127	sylib 221	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))
129		fuciso.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
130		fucinv.i	. . 3 ⊢ 𝐼 = (Inv‘𝑄)
131	129, 10, 82, 5, 16, 130, 4	fucinv 18039	. 2 ⊢ (𝜑 → (𝑈(𝐹𝐼𝐺)(𝑥 ∈ 𝐵 ↦ 𝑋) ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ (𝐺𝑁𝐹) ∧ ∀𝑦 ∈ 𝐵 (𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))))
132	1, 125, 128, 131	mpbir3and 1360	1 ⊢ (𝜑 → 𝑈(𝐹𝐼𝐺)(𝑥 ∈ 𝐵 ↦ 𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ∀wral 3078 Vcvv 3454 ⟨cop 4594 class class class wbr 5108 ↦ cmpt 5191 × cxp 5658 Rel wrel 5665 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 1^st c1st 7982 2^nd c2nd 7983 Xcixp 8893 Basecbs 17275 Hom chom 17327 compcco 17328 Catccat 17726 Idccid 17727 Sectcsect 17807 Invcinv 17808 Func cfunc 17917 Nat cnat 18007 FuncCat cfuc 18008

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-er 8692 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-fz 13542 df-struct 17213 df-slot 17248 df-ndx 17260 df-base 17276 df-hom 17340 df-cco 17341 df-cat 17730 df-cid 17731 df-sect 17810 df-inv 17811 df-func 17921 df-nat 18009 df-fuc 18010

This theorem is used by: fuciso 18041 yonedainv 18343

W3C validator