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Theorem invfuc 17995

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 2736	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		fucinv.j	. . . . . . . . . 10 ⊢ 𝐽 = (Inv‘𝐷)
5		fuciso.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6		funcrcl 17881	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8	7	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ Cat)
9	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
10		fuciso.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐶)
11		relfunc 17880	. . . . . . . . . . . . 13 ⊢ Rel (𝐶 Func 𝐷)
12		1st2ndbr 8046	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
13	11, 5, 12	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
14	10, 3, 13	funcf1 17884	. . . . . . . . . . 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 8046	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
18	11, 16, 17	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
19	10, 3, 18	funcf1 17884	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
20	19	ffvelcdmda 7079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
21		eqid 2736	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22	3, 4, 9, 15, 20, 21	invss 17779	. . . . . . . . 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 5167	. . . . . . . 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 5708	. . . . . . . 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 496	. . . . . . 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 6895	. . . . . 6 ⊢ 𝐵 ∈ V
30		mptelixpg 8954	. . . . . 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 234	. . . 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 7428	. . . . 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 8934	. . . 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 2845	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ X𝑦 ∈ 𝐵 (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)))
38		simpr2 1196	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ 𝐵)
39		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
40		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑋) = (𝑥 ∈ 𝐵 ↦ 𝑋)
41	40	fvmpt2 7002	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥))) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
42	39, 27, 41	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
43	2, 42	breqtrrd 5152	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 2899	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑈‘𝑧)
47		nfcv 2899	. . . . . . . . . . . . . . . 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 5171	. . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . 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 5138	. . . . . . . . . . . . . . 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 3594	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
59	14	adantr 480	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
64	3, 4, 58, 60, 62, 63	isinv 17778	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 494	. . . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (comp‘𝐷) = (comp‘𝐷)
68		eqid 2736	. . . . . . . . . . . 12 ⊢ (Id‘𝐷) = (Id‘𝐷)
69	3, 21, 67, 68, 63, 58, 60, 62	issect 17771	. . . . . . . . . . 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 232	. . . . . . . . . 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 1144	. . . . . . . . 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 7425	. . . . . . . 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 1195	. . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
76	13	adantr 480	. . . . . . . . . . 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 17886	. . . . . . . . . 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 1197	. . . . . . . . . 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 17700	. . . . . . . 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 2772	. . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (𝐹𝑁𝐺))
84	82, 83	nat1st2nd 17972	. . . . . . . . 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 17974	. . . . . . . 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 1143	. . . . . . . 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 17703	. . . . . . 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 17976	. . . . . . . 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 7426	. . . . . . 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 2775	. . . . . 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 7425	. . . . 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 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑈‘𝑦)
94		nfcv 2899	. . . . . . . . . . . . 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 5171	. . . . . . . . . . . 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 7428	. . . . . . . . . . . . 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 5138	. . . . . . . . . . . 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 3594	. . . . . . . . . . 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 17778	. . . . . . . . . 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 232	. . . . . . . . 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 495	. . . . . . . 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 17771	. . . . . . . 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 232	. . . . . . 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 1142	. . . . . 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 1143	. . . . . . 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 480	. . . . . . . . 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 17886	. . . . . . . 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 17702	. . . . . 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 17703	. . . . 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 17974	. . . . . . . 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 17703	. . . . . . 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 1144	. . . . . . . 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 7426	. . . . . . 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 17701	. . . . . . 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 2775	. . . . . 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 7426	. . . . 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 2776	. . . 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 3191	. . 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 17969	. . 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 713	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ (𝐺𝑁𝐹))
126		nfv 1914	. . . 4 ⊢ Ⅎ𝑦(𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥)
127	126, 96, 100	cbvralw 3290	. . 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 218	. 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 17994	. 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 1343	1 ⊢ (𝜑 → 𝑈(𝐹𝐼𝐺)(𝑥 ∈ 𝐵 ↦ 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3052 Vcvv 3464 ⟨cop 4612 class class class wbr 5124 ↦ cmpt 5206 × cxp 5657 Rel wrel 5664 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 Xcixp 8916 Basecbs 17233 Hom chom 17287 compcco 17288 Catccat 17681 Idccid 17682 Sectcsect 17762 Invcinv 17763 Func cfunc 17872 Nat cnat 17962 FuncCat cfuc 17963

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-struct 17171 df-slot 17206 df-ndx 17218 df-base 17234 df-hom 17300 df-cco 17301 df-cat 17685 df-cid 17686 df-sect 17765 df-inv 17766 df-func 17876 df-nat 17964 df-fuc 17965

This theorem is referenced by: fuciso 17996 yonedainv 18298

W3C validator