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Theorem invfuc 18152

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 2760	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		fucinv.j	. . . . . . . . . 10 ⊢ 𝐽 = (Inv‘𝐷)
5		fuciso.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6		funcrcl 18038	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
7	5, 6	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8	7	simprd 501	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ Cat)
9	8	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
10		fuciso.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐶)
11		relfunc 18037	. . . . . . . . . . . . 13 ⊢ Rel (𝐶 Func 𝐷)
12		1st2ndbr 8053	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
13	11, 5, 12	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
14	10, 3, 13	funcf1 18041	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
15	14	ffvelcdmda 7084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
16		fuciso.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
17		1st2ndbr 8053	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
18	11, 16, 17	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
19	10, 3, 18	funcf1 18041	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
20	19	ffvelcdmda 7084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
21		eqid 2760	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22	3, 4, 9, 15, 20, 21	invss 17936	. . . . . . . . 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 5147	. . . . . . . 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 5699	. . . . . . . 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 503	. . . . . . 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 3154	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
29	10	fvexi 6899	. . . . . 6 ⊢ 𝐵 ∈ V
30		mptelixpg 8963	. . . . . 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 6885	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑦))
34		fveq2 6885	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑦))
35	33, 34	oveq12d 7438	. . . . 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 8943	. . . 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 2870	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ X𝑦 ∈ 𝐵 (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)))
38		simpr2 1214	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ 𝐵)
39		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
40		eqid 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑋) = (𝑥 ∈ 𝐵 ↦ 𝑋)
41	40	fvmpt2 7005	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥))) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
42	39, 27, 41	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
43	2, 42	breqtrrd 5132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥))
44	43	ralrimiva 3154	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥))
45	44	adantr 486	. . . . . . . . . . . . . 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 2922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑈‘𝑧)
47		nfcv 2922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))
48		nffvmpt1 6896	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)
49	46, 47, 48	nfbr 5151	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)
50		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑈‘𝑥) = (𝑈‘𝑧))
51		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑧))
52		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑧))
53	51, 52	oveq12d 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) = (((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧)))
54		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧))
55	50, 53, 54	breq123d 5116	. . . . . . . . . . . . . . 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 3564	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
59	14	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
60	59, 38	ffvelcdmd 7085	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐹)‘𝑧) ∈ (Base‘𝐷))
61	19	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
62	61, 38	ffvelcdmd 7085	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐺)‘𝑧) ∈ (Base‘𝐷))
63		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
64	3, 4, 58, 60, 62, 63	isinv 17935	. . . . . . . . . . . . 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 500	. . . . . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ (comp‘𝐷) = (comp‘𝐷)
68		eqid 2760	. . . . . . . . . . . 12 ⊢ (Id‘𝐷) = (Id‘𝐷)
69	3, 21, 67, 68, 63, 58, 60, 62	issect 17928	. . . . . . . . . . 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 1162	. . . . . . . . 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 7435	. . . . . . . 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 1213	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ 𝐵)
74	59, 73	ffvelcdmd 7085	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
75		eqid 2760	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
76	13	adantr 486	. . . . . . . . . . 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 18043	. . . . . . . . . 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 1215	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
79	77, 78	ffvelcdmd 7085	. . . . . . . . 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 17857	. . . . . . . 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 2796	. . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (𝐹𝑁𝐺))
84	82, 83	nat1st2nd 18129	. . . . . . . . 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 18131	. . . . . . . 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 1161	. . . . . . . 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 17860	. . . . . . 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 18133	. . . . . . . 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 7436	. . . . . . 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 2799	. . . . . 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 7435	. . . . 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 7085	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
93		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑈‘𝑦)
94		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))
95		nffvmpt1 6896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)
96	93, 94, 95	nfbr 5151	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)
97		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑈‘𝑥) = (𝑈‘𝑦))
98	34, 33	oveq12d 7438	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) = (((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦)))
99		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))
100	97, 98, 99	breq123d 5116	. . . . . . . . . . . 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 3564	. . . . . . . . . . 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 17935	. . . . . . . . . 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 501	. . . . . . . 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 17928	. . . . . . . 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 1160	. . . . . 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 1161	. . . . . . 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 486	. . . . . . . . 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 18043	. . . . . . . 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 7085	. . . . . . 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 17859	. . . . . 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 17860	. . . . 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 18131	. . . . . . . 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 17860	. . . . . . 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 1162	. . . . . . . 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 7436	. . . . . . 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 17858	. . . . . . 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 2799	. . . . . 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 7436	. . . . 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 2800	. . . 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 3208	. . 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 18126	. . 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 726	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ (𝐺𝑁𝐹))
126		nfv 1946	. . . 4 ⊢ Ⅎ𝑦(𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥)
127	126, 96, 100	cbvralw 3304	. . 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 18151	. 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 1361	1 ⊢ (𝜑 → 𝑈(𝐹𝐼𝐺)(𝑥 ∈ 𝐵 ↦ 𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 Vcvv 3450 ⟨cop 4589 class class class wbr 5102 ↦ cmpt 5185 × cxp 5648 Rel wrel 5655 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 Xcixp 8925 Basecbs 17387 Hom chom 17439 compcco 17440 Catccat 17838 Idccid 17839 Sectcsect 17919 Invcinv 17920 Func cfunc 18029 Nat cnat 18119 FuncCat cfuc 18120

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-fz 13640 df-struct 17325 df-slot 17360 df-ndx 17372 df-base 17388 df-hom 17452 df-cco 17453 df-cat 17842 df-cid 17843 df-sect 17922 df-inv 17923 df-func 18033 df-nat 18121 df-fuc 18122

This theorem is used by: fuciso 18153 yonedainv 18455

W3C validator