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Theorem invfuc 16457

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 2610	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		fucinv.j	. . . . . . . . . 10 ⊢ 𝐽 = (Inv‘𝐷)
5		fuciso.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
6		funcrcl 16346	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8	7	simprd 478	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ Cat)
9	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
10		fuciso.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐶)
11		relfunc 16345	. . . . . . . . . . . . 13 ⊢ Rel (𝐶 Func 𝐷)
12		1st2ndbr 7108	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
13	11, 5, 12	sylancr 694	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
14	10, 3, 13	funcf1 16349	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
15	14	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
16		fuciso.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
17		1st2ndbr 7108	. . . . . . . . . . . . 13 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
18	11, 16, 17	sylancr 694	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
19	10, 3, 18	funcf1 16349	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
20	19	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
21		eqid 2610	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22	3, 4, 9, 15, 20, 21	invss 16244	. . . . . . . . 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 4626	. . . . . . . 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 5071	. . . . . . . 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 479	. . . . . . 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 2949	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
29		fvex 6113	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
30	10, 29	eqeltri 2684	. . . . . 6 ⊢ 𝐵 ∈ V
31		mptelixpg 7831	. . . . . 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))))
32	30, 31	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)))
33	28, 32	sylibr 223	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
34		fveq2 6103	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑦))
35		fveq2 6103	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑦))
36	34, 35	oveq12d 6567	. . . . 5 ⊢ (𝑥 = 𝑦 → (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)) = (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)))
37	36	cbvixpv 7812	. . . 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))
38	33, 37	syl6eleq 2698	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ X𝑦 ∈ 𝐵 (((1^st ‘𝐺)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑦)))
39		simpr2 1061	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ 𝐵)
40		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
41		eqid 2610	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑋) = (𝑥 ∈ 𝐵 ↦ 𝑋)
42	41	fvmpt2 6200	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑋 ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥))) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
43	40, 27, 42	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
44	2, 43	breqtrrd 4611	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥))
45	44	ralrimiva 2949	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥))
46	45	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))
47		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑈‘𝑧)
48		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))
49		nffvmpt1 6111	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)
50	47, 48, 49	nfbr 4629	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑈‘𝑧)(((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧)
51		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑈‘𝑥) = (𝑈‘𝑧))
52		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑧))
53		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑧))
54	52, 53	oveq12d 6567	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) = (((1^st ‘𝐹)‘𝑧)𝐽((1^st ‘𝐺)‘𝑧)))
55		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑧))
56	51, 54, 55	breq123d 4597	. . . . . . . . . . . . . . 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)))
57	50, 56	rspc 3276	. . . . . . . . . . . . . 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)))
58	39, 46, 57	sylc 63	. . . . . . . . . . . . 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))
59	8	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
60	14	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
61	60, 39	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐹)‘𝑧) ∈ (Base‘𝐷))
62	19	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
63	62, 39	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐺)‘𝑧) ∈ (Base‘𝐷))
64		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
65	3, 4, 59, 61, 63, 64	isinv 16243	. . . . . . . . . . . . 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))))
66	58, 65	mpbid 221	. . . . . . . . . . . 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)))
67	66	simpld 474	. . . . . . . . . . 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))
68		eqid 2610	. . . . . . . . . . . 12 ⊢ (comp‘𝐷) = (comp‘𝐷)
69		eqid 2610	. . . . . . . . . . . 12 ⊢ (Id‘𝐷) = (Id‘𝐷)
70	3, 21, 68, 69, 64, 59, 61, 63	issect 16236	. . . . . . . . . . 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)))))
71	67, 70	mpbid 221	. . . . . . . . . 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))))
72	71	simp3d 1068	. . . . . . . . 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)))
73	72	oveq1d 6564	. . . . . . . 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)))
74		simpr1 1060	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ 𝐵)
75	60, 74	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
76		eqid 2610	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
77	13	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
78	10, 76, 21, 77, 74, 39	funcf2 16351	. . . . . . . . . 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)))
79		simpr3 1062	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
80	78, 79	ffvelrnd 6268	. . . . . . . . 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)))
81	3, 21, 69, 59, 75, 68, 61, 80	catlid 16167	. . . . . . . 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))
82	73, 81	eqtr2d 2645	. . . . . . 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)))
83		fuciso.n	. . . . . . . . 9 ⊢ 𝑁 = (𝐶 Nat 𝐷)
84	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (𝐹𝑁𝐺))
85	83, 84	nat1st2nd 16434	. . . . . . . . 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)⟩))
86	83, 85, 10, 21, 39	natcl 16436	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈‘𝑧) ∈ (((1^st ‘𝐹)‘𝑧)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑧)))
87	71	simp2d 1067	. . . . . . . 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)))
88	3, 21, 68, 59, 75, 61, 63, 80, 86, 61, 87	catass 16170	. . . . . . 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))))
89	83, 85, 10, 76, 68, 74, 39, 79	nati 16438	. . . . . . . 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)))
90	89	oveq2d 6565	. . . . . . 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))))
91	82, 88, 90	3eqtrd 2648	. . . . . 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))))
92	91	oveq1d 6564	. . . . 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)))
93	62, 74	ffvelrnd 6268	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1^st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
94		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑈‘𝑦)
95		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))
96		nffvmpt1 6111	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)
97	94, 95, 96	nfbr 4629	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦)
98		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑈‘𝑥) = (𝑈‘𝑦))
99	35, 34	oveq12d 6567	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) = (((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦)))
100		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))
101	98, 99, 100	breq123d 4597	. . . . . . . . . . . 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)))
102	97, 101	rspc 3276	. . . . . . . . . . 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)))
103	74, 46, 102	sylc 63	. . . . . . . . . 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))
104	3, 4, 59, 75, 93, 64	isinv 16243	. . . . . . . . . 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))))
105	103, 104	mpbid 221	. . . . . . . . 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)))
106	105	simprd 478	. . . . . . . 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))
107	3, 21, 68, 69, 64, 59, 93, 75	issect 16236	. . . . . . . 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)))))
108	106, 107	mpbid 221	. . . . . . 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))))
109	108	simp1d 1066	. . . . . 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)))
110	108	simp2d 1067	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈‘𝑦) ∈ (((1^st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑦)))
111	18	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
112	10, 76, 21, 111, 74, 39	funcf2 16351	. . . . . . . 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)))
113	112, 79	ffvelrnd 6268	. . . . . . 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)))
114	3, 21, 68, 59, 75, 93, 63, 110, 113	catcocl 16169	. . . . . 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)))
115	3, 21, 68, 59, 93, 75, 63, 109, 114, 61, 87	catass 16170	. . . . 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))))
116	83, 85, 10, 21, 74	natcl 16436	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈‘𝑦) ∈ (((1^st ‘𝐹)‘𝑦)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑦)))
117	3, 21, 68, 59, 93, 75, 93, 109, 116, 63, 113	catass 16170	. . . . . . 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))))
118	108	simp3d 1068	. . . . . . . 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)))
119	118	oveq2d 6565	. . . . . . 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))))
120	3, 21, 69, 59, 93, 68, 63, 113	catrid 16168	. . . . . . 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))
121	117, 119, 120	3eqtrd 2648	. . . . . 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))
122	121	oveq2d 6565	. . . . 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)))
123	92, 115, 122	3eqtrrd 2649	. . . 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)))
124	123	ralrimivvva 2955	. . 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)))
125	83, 10, 76, 21, 68, 16, 5	isnat2 16431	. . 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)))))
126	38, 124, 125	mpbir2and 959	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝑋) ∈ (𝐺𝑁𝐹))
127		nfv 1830	. . . 4 ⊢ Ⅎ𝑦(𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥)
128	127, 97, 101	cbvral 3143	. . 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))
129	45, 128	sylib 207	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑈‘𝑦)(((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑦))
130		fuciso.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
131		fucinv.i	. . 3 ⊢ 𝐼 = (Inv‘𝑄)
132	130, 10, 83, 5, 16, 131, 4	fucinv 16456	. 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))))
133	1, 126, 129, 132	mpbir3and 1238	1 ⊢ (𝜑 → 𝑈(𝐹𝐼𝐺)(𝑥 ∈ 𝐵 ↦ 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 Rel wrel 5043 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 Xcixp 7794 Basecbs 15695 Hom chom 15779 compcco 15780 Catccat 16148 Idccid 16149 Sectcsect 16227 Invcinv 16228 Func cfunc 16337 Nat cnat 16424 FuncCat cfuc 16425

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-hom 15793 df-cco 15794 df-cat 16152 df-cid 16153 df-sect 16230 df-inv 16231 df-func 16341 df-nat 16426 df-fuc 16427

This theorem is referenced by: fuciso 16458 yonedainv 16744

W3C validator