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Theorem invfuc 17939

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 2741	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		fucinv.j	. . . . . . . . . 10 ⊢ 𝐽 = (Inv‘𝐷)
5		fuciso.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6		funcrcl 17825	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8	7	simprd 497	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ Cat)
9	8	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
10		fuciso.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐶)
11		relfunc 17824	. . . . . . . . . . . . 13 ⊢ Rel (𝐶 Func 𝐷)
12		1st2ndbr 7986	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
13	11, 5, 12	sylancr 594	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
14	10, 3, 13	funcf1 17828	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
15	14	ffvelcdmda 7028	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
16		fuciso.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
17		1st2ndbr 7986	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
18	11, 16, 17	sylancr 594	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
19	10, 3, 18	funcf1 17828	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
20	19	ffvelcdmda 7028	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
21		eqid 2741	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22	3, 4, 9, 15, 20, 21	invss 17723	. . . . . . . . 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 5117	. . . . . . . 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 15	. . . . . . 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 5669	. . . . . . . 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 499	. . . . . . 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 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
28	27	ralrimiva 3133	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
29	10	fvexi 6844	. . . . . 6 ⊢ 𝐵 ∈ V
30		mptelixpg 8877	. . . . . 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 236	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
33		fveq2 6830	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑦))
34		fveq2 6830	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑦))
35	33, 34	oveq12d 7377	. . . . 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 8857	. . . 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 2851	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ X𝑦 ∈ 𝐵 (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)))
38		simpr2 1203	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ 𝐵)
39		simpr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
40		eqid 2741	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑋) = (𝑥 ∈ 𝐵 ↦ 𝑋)
41	40	fvmpt2 6950	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥))) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
42	39, 27, 41	syl2anc 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
43	2, 42	breqtrrd 5102	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥))
44	43	ralrimiva 3133	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥))
45	44	adantr 482	. . . . . . . . . . . . . 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 2903	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑈‘𝑧)
47		nfcv 2903	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))
48		nffvmpt1 6841	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)
49	46, 47, 48	nfbr 5121	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)
50		fveq2 6830	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑈‘𝑥) = (𝑈‘𝑧))
51		fveq2 6830	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑧))
52		fveq2 6830	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑧))
53	51, 52	oveq12d 7377	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) = (((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧)))
54		fveq2 6830	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧))
55	50, 53, 54	breq123d 5088	. . . . . . . . . . . . . . 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 3549	. . . . . . . . . . . . . 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 65	. . . . . . . . . . . . 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 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
59	14	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
60	59, 38	ffvelcdmd 7029	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐹)‘𝑧) ∈ (Base‘𝐷))
61	19	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
62	61, 38	ffvelcdmd 7029	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐺)‘𝑧) ∈ (Base‘𝐷))
63		eqid 2741	. . . . . . . . . . . . . 14 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
64	3, 4, 58, 60, 62, 63	isinv 17722	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 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 496	. . . . . . . . . . 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 2741	. . . . . . . . . . . 12 ⊢ (comp‘𝐷) = (comp‘𝐷)
68		eqid 2741	. . . . . . . . . . . 12 ⊢ (Id‘𝐷) = (Id‘𝐷)
69	3, 21, 67, 68, 63, 58, 60, 62	issect 17715	. . . . . . . . . . 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 234	. . . . . . . . . 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 1151	. . . . . . . . 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 7374	. . . . . . . 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 1202	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ 𝐵)
74	59, 73	ffvelcdmd 7029	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
75		eqid 2741	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
76	13	adantr 482	. . . . . . . . . . 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 17830	. . . . . . . . . 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 1204	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
79	77, 78	ffvelcdmd 7029	. . . . . . . . 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 17644	. . . . . . . 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 2777	. . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (𝐹𝑁𝐺))
84	82, 83	nat1st2nd 17916	. . . . . . . . 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 17918	. . . . . . . 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 1150	. . . . . . . 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 17647	. . . . . . 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 17920	. . . . . . . 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 7375	. . . . . . 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 2780	. . . . . 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 7374	. . . . 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 7029	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
93		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑈‘𝑦)
94		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))
95		nffvmpt1 6841	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)
96	93, 94, 95	nfbr 5121	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)
97		fveq2 6830	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑈‘𝑥) = (𝑈‘𝑦))
98	34, 33	oveq12d 7377	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) = (((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦)))
99		fveq2 6830	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))
100	97, 98, 99	breq123d 5088	. . . . . . . . . . . 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 3549	. . . . . . . . . . 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 65	. . . . . . . . . 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 17722	. . . . . . . . . 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 234	. . . . . . . . 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 497	. . . . . . . 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 17715	. . . . . . . 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 234	. . . . . . 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 1149	. . . . . 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 1150	. . . . . . 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 482	. . . . . . . . 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 17830	. . . . . . . 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 7029	. . . . . . 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 17646	. . . . . 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 17647	. . . . 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 17918	. . . . . . . 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 17647	. . . . . . 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 1151	. . . . . . . 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 7375	. . . . . . 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 17645	. . . . . . 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 2780	. . . . . 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 7375	. . . . 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 2781	. . . 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 3187	. . 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 17913	. . 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 720	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ (𝐺𝑁𝐹))
126		nfv 1922	. . . 4 ⊢ Ⅎ𝑦(𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥)
127	126, 96, 100	cbvralw 3283	. . 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 220	. 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 17938	. 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 1350	1 ⊢ (𝜑 → 𝑈(𝐹𝐼𝐺)(𝑥 ∈ 𝐵 ↦ 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ∀wral 3055 Vcvv 3433 ⟨cop 4563 class class class wbr 5074 ↦ cmpt 5155 × cxp 5618 Rel wrel 5625 ⟶wf 6484 ‘cfv 6488 (class class class)co 7359 1^st c1st 7931 2^nd c2nd 7932 Xcixp 8839 Basecbs 17174 Hom chom 17226 compcco 17227 Catccat 17625 Idccid 17626 Sectcsect 17706 Invcinv 17707 Func cfunc 17816 Nat cnat 17906 FuncCat cfuc 17907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-om 7810 df-1st 7933 df-2nd 7934 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-struct 17112 df-slot 17147 df-ndx 17159 df-base 17175 df-hom 17239 df-cco 17240 df-cat 17629 df-cid 17630 df-sect 17709 df-inv 17710 df-func 17820 df-nat 17908 df-fuc 17909

This theorem is referenced by: fuciso 17940 yonedainv 18242

W3C validator