Theorem uncfcurf 17491

Description: Cancellation of uncurry with curry. (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
uncfcurf.g	⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
uncfcurf.c	⊢ (𝜑 → 𝐶 ∈ Cat)
uncfcurf.d	⊢ (𝜑 → 𝐷 ∈ Cat)
uncfcurf.f	⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))

Assertion

Ref	Expression
uncfcurf	⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = 𝐹)

Proof of Theorem uncfcurf

Dummy variables 𝑓 𝑔 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2823	. . . . . . 7 ⊢ (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
2		uncfcurf.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
3	2	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
4		uncfcurf.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
5		funcrcl 17135	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸) → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
6	4, 5	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
7	6	simprd 498	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
8	7	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐸 ∈ Cat)
9		uncfcurf.g	. . . . . . . . 9 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
10		eqid 2823	. . . . . . . . 9 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
11		uncfcurf.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	9, 10, 11, 2, 4	curfcl 17484	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
13	12	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
14		eqid 2823	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
15		eqid 2823	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		simprl 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶))
17		simprr 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
18	1, 3, 8, 13, 14, 15, 16, 17	uncf1 17488	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦))
19	11	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐶 ∈ Cat)
20	4	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
21		eqid 2823	. . . . . . 7 ⊢ ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑥)
22	9, 14, 19, 3, 20, 15, 16, 21, 17	curf11 17478	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦) = (𝑥(1st ‘𝐹)𝑦))
23	18, 22	eqtrd 2858	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦))
24	23	ralrimivva 3193	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦))
25		eqid 2823	. . . . . . . 8 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
26	25, 14, 15	xpcbas 17430	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×c 𝐷))
27		eqid 2823	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
28		relfunc 17134	. . . . . . . 8 ⊢ Rel ((𝐶 ×c 𝐷) Func 𝐸)
29	1, 2, 7, 12	uncfcl 17487	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) ∈ ((𝐶 ×c 𝐷) Func 𝐸))
30		1st2ndbr 7743	. . . . . . . 8 ⊢ ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)))
31	28, 29, 30	sylancr 589	. . . . . . 7 ⊢ (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)))
32	26, 27, 31	funcf1 17138	. . . . . 6 ⊢ (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸))
33	32	ffnd 6517	. . . . 5 ⊢ (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷)))
34		1st2ndbr 7743	. . . . . . . 8 ⊢ ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
35	28, 4, 34	sylancr 589	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
36	26, 27, 35	funcf1 17138	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸))
37	36	ffnd 6517	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐷)))
38		eqfnov2 7283	. . . . 5 ⊢ (((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷)) ∧ (1st ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐷))) → ((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (1st ‘𝐹) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦)))
39	33, 37, 38	syl2anc 586	. . . 4 ⊢ (𝜑 → ((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (1st ‘𝐹) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦)))
40	24, 39	mpbird 259	. . 3 ⊢ (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (1st ‘𝐹))
41	2	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat)
42	7	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝐸 ∈ Cat)
43	12	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
44	16	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶))
45	44	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑥 ∈ (Base‘𝐶))
46	17	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
47	46	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑦 ∈ (Base‘𝐷))
48		eqid 2823	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
49		eqid 2823	. . . . . . . . . . 11 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
50		simprl 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐶))
51	50	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑧 ∈ (Base‘𝐶))
52		simprr 771	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑤 ∈ (Base‘𝐷))
53	52	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑤 ∈ (Base‘𝐷))
54		simprl 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧))
55		simprr 771	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))
56	1, 41, 42, 43, 14, 15, 45, 47, 48, 49, 51, 53, 54, 55	uncf2 17489	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (𝑓(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩)𝑔) = ((((𝑥(2nd ‘𝐺)𝑧)‘𝑓)‘𝑤)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤))((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑤)‘𝑔)))
57	11	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat)
58	4	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
59	9, 14, 57, 41, 58, 15, 45, 21, 47	curf11 17478	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦) = (𝑥(1st ‘𝐹)𝑦))
60		df-ov 7161	. . . . . . . . . . . . . . 15 ⊢ (𝑥(1st ‘𝐹)𝑦) = ((1st ‘𝐹)‘⟨𝑥, 𝑦⟩)
61	59, 60	syl6eq 2874	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦) = ((1st ‘𝐹)‘⟨𝑥, 𝑦⟩))
62	9, 14, 57, 41, 58, 15, 45, 21, 53	curf11 17478	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤) = (𝑥(1st ‘𝐹)𝑤))
63		df-ov 7161	. . . . . . . . . . . . . . 15 ⊢ (𝑥(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘⟨𝑥, 𝑤⟩)
64	62, 63	syl6eq 2874	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤) = ((1st ‘𝐹)‘⟨𝑥, 𝑤⟩))
65	61, 64	opeq12d 4813	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤)⟩ = ⟨((1st ‘𝐹)‘⟨𝑥, 𝑦⟩), ((1st ‘𝐹)‘⟨𝑥, 𝑤⟩)⟩)
66		eqid 2823	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝐺)‘𝑧) = ((1st ‘𝐺)‘𝑧)
67	9, 14, 57, 41, 58, 15, 51, 66, 53	curf11 17478	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤) = (𝑧(1st ‘𝐹)𝑤))
68		df-ov 7161	. . . . . . . . . . . . . 14 ⊢ (𝑧(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘⟨𝑧, 𝑤⟩)
69	67, 68	syl6eq 2874	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤) = ((1st ‘𝐹)‘⟨𝑧, 𝑤⟩))
70	65, 69	oveq12d 7176	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤)) = (⟨((1st ‘𝐹)‘⟨𝑥, 𝑦⟩), ((1st ‘𝐹)‘⟨𝑥, 𝑤⟩)⟩(comp‘𝐸)((1st ‘𝐹)‘⟨𝑧, 𝑤⟩)))
71		eqid 2823	. . . . . . . . . . . . . 14 ⊢ (Id‘𝐷) = (Id‘𝐷)
72		eqid 2823	. . . . . . . . . . . . . 14 ⊢ ((𝑥(2nd ‘𝐺)𝑧)‘𝑓) = ((𝑥(2nd ‘𝐺)𝑧)‘𝑓)
73	9, 14, 57, 41, 58, 15, 48, 71, 45, 51, 54, 72, 53	curf2val 17482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (((𝑥(2nd ‘𝐺)𝑧)‘𝑓)‘𝑤) = (𝑓(⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)))
74		df-ov 7161	. . . . . . . . . . . . 13 ⊢ (𝑓(⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)
75	73, 74	syl6eq 2874	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (((𝑥(2nd ‘𝐺)𝑧)‘𝑓)‘𝑤) = ((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩))
76		eqid 2823	. . . . . . . . . . . . . 14 ⊢ (Id‘𝐶) = (Id‘𝐶)
77	9, 14, 57, 41, 58, 15, 45, 21, 47, 49, 76, 53, 55	curf12 17479	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑤)‘𝑔) = (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑤⟩)𝑔))
78		df-ov 7161	. . . . . . . . . . . . 13 ⊢ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑤⟩)𝑔) = ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑔⟩)
79	77, 78	syl6eq 2874	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑤)‘𝑔) = ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑔⟩))
80	70, 75, 79	oveq123d 7179	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((((𝑥(2nd ‘𝐺)𝑧)‘𝑓)‘𝑤)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤))((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑤)‘𝑔)) = (((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)(⟨((1st ‘𝐹)‘⟨𝑥, 𝑦⟩), ((1st ‘𝐹)‘⟨𝑥, 𝑤⟩)⟩(comp‘𝐸)((1st ‘𝐹)‘⟨𝑧, 𝑤⟩))((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑔⟩)))
81		eqid 2823	. . . . . . . . . . . 12 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
82		eqid 2823	. . . . . . . . . . . 12 ⊢ (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
83		eqid 2823	. . . . . . . . . . . 12 ⊢ (comp‘𝐸) = (comp‘𝐸)
84	35	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
85	84	adantr 483	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
86		opelxpi 5594	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
87	86	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
88	87	adantr 483	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
89	45, 53	opelxpd 5595	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ⟨𝑥, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
90		opelxpi 5594	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
91	90	adantl 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
92	91	adantr 483	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
93	14, 48, 76, 57, 45	catidcl 16955	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
94	93, 55	opelxpd 5595	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑥), 𝑔⟩ ∈ ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑤)))
95	25, 14, 15, 48, 49, 45, 47, 45, 53, 81	xpchom2 17438	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑥, 𝑤⟩) = ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑤)))
96	94, 95	eleqtrrd 2918	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑥), 𝑔⟩ ∈ (⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑥, 𝑤⟩))
97	15, 49, 71, 41, 53	catidcl 16955	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((Id‘𝐷)‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤))
98	54, 97	opelxpd 5595	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ⟨𝑓, ((Id‘𝐷)‘𝑤)⟩ ∈ ((𝑥(Hom ‘𝐶)𝑧) × (𝑤(Hom ‘𝐷)𝑤)))
99	25, 14, 15, 48, 49, 45, 53, 51, 53, 81	xpchom2 17438	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (⟨𝑥, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩) = ((𝑥(Hom ‘𝐶)𝑧) × (𝑤(Hom ‘𝐷)𝑤)))
100	98, 99	eleqtrrd 2918	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ⟨𝑓, ((Id‘𝐷)‘𝑤)⟩ ∈ (⟨𝑥, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩))
101	26, 81, 82, 83, 85, 88, 89, 92, 96, 100	funcco 17143	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑓, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑦⟩, ⟨𝑥, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨((Id‘𝐶)‘𝑥), 𝑔⟩)) = (((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)(⟨((1st ‘𝐹)‘⟨𝑥, 𝑦⟩), ((1st ‘𝐹)‘⟨𝑥, 𝑤⟩)⟩(comp‘𝐸)((1st ‘𝐹)‘⟨𝑧, 𝑤⟩))((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑔⟩)))
102		eqid 2823	. . . . . . . . . . . . . . 15 ⊢ (comp‘𝐶) = (comp‘𝐶)
103		eqid 2823	. . . . . . . . . . . . . . 15 ⊢ (comp‘𝐷) = (comp‘𝐷)
104	25, 14, 15, 48, 49, 45, 47, 45, 53, 102, 103, 82, 51, 53, 93, 55, 54, 97	xpcco2 17439	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (⟨𝑓, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑦⟩, ⟨𝑥, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨((Id‘𝐶)‘𝑥), 𝑔⟩) = ⟨(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥)), (((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔)⟩)
105	104	fveq2d 6676	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑓, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑦⟩, ⟨𝑥, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨((Id‘𝐶)‘𝑥), 𝑔⟩)) = ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥)), (((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔)⟩))
106		df-ov 7161	. . . . . . . . . . . . 13 ⊢ ((𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥))(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)(((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔)) = ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥)), (((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔)⟩)
107	105, 106	syl6eqr 2876	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑓, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑦⟩, ⟨𝑥, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨((Id‘𝐶)‘𝑥), 𝑔⟩)) = ((𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥))(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)(((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔)))
108	14, 48, 76, 57, 45, 102, 51, 54	catrid 16957	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥)) = 𝑓)
109	15, 49, 71, 41, 47, 103, 53, 55	catlid 16956	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔) = 𝑔)
110	108, 109	oveq12d 7176	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥))(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)(((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔)) = (𝑓(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)𝑔))
111	107, 110	eqtrd 2858	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑓, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑦⟩, ⟨𝑥, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨((Id‘𝐶)‘𝑥), 𝑔⟩)) = (𝑓(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)𝑔))
112	80, 101, 111	3eqtr2d 2864	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((((𝑥(2nd ‘𝐺)𝑧)‘𝑓)‘𝑤)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑧))‘𝑤))((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑤)‘𝑔)) = (𝑓(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)𝑔))
113	56, 112	eqtrd 2858	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (𝑓(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩)𝑔) = (𝑓(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)𝑔))
114	113	ralrimivva 3193	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤)(𝑓(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩)𝑔) = (𝑓(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)𝑔))
115		eqid 2823	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
116	31	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)))
117	26, 81, 115, 116, 87, 91	funcf2 17140	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩):(⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟶(((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))‘⟨𝑧, 𝑤⟩)))
118	25, 14, 15, 48, 49, 44, 46, 50, 52, 81	xpchom2 17438	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩) = ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤)))
119	118	feq2d 6502	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ((⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩):(⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟶(((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))‘⟨𝑧, 𝑤⟩)) ↔ (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))‘⟨𝑧, 𝑤⟩))))
120	117, 119	mpbid 234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))‘⟨𝑧, 𝑤⟩)))
121	120	ffnd 6517	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤)))
122	26, 81, 115, 84, 87, 91	funcf2 17140	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩):(⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟶(((1st ‘𝐹)‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st ‘𝐹)‘⟨𝑧, 𝑤⟩)))
123	118	feq2d 6502	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩):(⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟶(((1st ‘𝐹)‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st ‘𝐹)‘⟨𝑧, 𝑤⟩)) ↔ (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st ‘𝐹)‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st ‘𝐹)‘⟨𝑧, 𝑤⟩))))
124	122, 123	mpbid 234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st ‘𝐹)‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st ‘𝐹)‘⟨𝑧, 𝑤⟩)))
125	124	ffnd 6517	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤)))
126		eqfnov2 7283	. . . . . . . . 9 ⊢ (((⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤)) ∧ (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))) → ((⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤)(𝑓(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩)𝑔) = (𝑓(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)𝑔)))
127	121, 125, 126	syl2anc 586	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ((⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤)(𝑓(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩)𝑔) = (𝑓(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)𝑔)))
128	114, 127	mpbird 259	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩))
129	128	ralrimivva 3193	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩))
130	129	ralrimivva 3193	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩))
131		oveq2 7166	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩))
132		oveq2 7166	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑢(2nd ‘𝐹)𝑣) = (𝑢(2nd ‘𝐹)⟨𝑧, 𝑤⟩))
133	131, 132	eqeq12d 2839	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣) ↔ (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (𝑢(2nd ‘𝐹)⟨𝑧, 𝑤⟩)))
134	133	ralxp 5714	. . . . . . 7 ⊢ (∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (𝑢(2nd ‘𝐹)⟨𝑧, 𝑤⟩))
135		oveq1 7165	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩))
136		oveq1 7165	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢(2nd ‘𝐹)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩))
137	135, 136	eqeq12d 2839	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (𝑢(2nd ‘𝐹)⟨𝑧, 𝑤⟩) ↔ (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)))
138	137	2ralbidv 3201	. . . . . . 7 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (𝑢(2nd ‘𝐹)⟨𝑧, 𝑤⟩) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)))
139	134, 138	syl5bb 285	. . . . . 6 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)))
140	139	ralxp 5714	. . . . 5 ⊢ (∀𝑢 ∈ ((Base‘𝐶) × (Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩))
141	130, 140	sylibr 236	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ ((Base‘𝐶) × (Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣))
142	26, 31	funcfn2 17141	. . . . 5 ⊢ (𝜑 → (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
143	26, 35	funcfn2 17141	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
144		eqfnov2 7283	. . . . 5 ⊢ (((2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))) ∧ (2nd ‘𝐹) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))) → ((2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (2nd ‘𝐹) ↔ ∀𝑢 ∈ ((Base‘𝐶) × (Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣)))
145	142, 143, 144	syl2anc 586	. . . 4 ⊢ (𝜑 → ((2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (2nd ‘𝐹) ↔ ∀𝑢 ∈ ((Base‘𝐶) × (Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣)))
146	141, 145	mpbird 259	. . 3 ⊢ (𝜑 → (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (2nd ‘𝐹))
147	40, 146	opeq12d 4813	. 2 ⊢ (𝜑 → ⟨(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)), (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟩ = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
148		1st2nd 7740	. . 3 ⊢ ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = ⟨(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)), (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟩)
149	28, 29, 148	sylancr 589	. 2 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = ⟨(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)), (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟩)
150		1st2nd 7740	. . 3 ⊢ ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
151	28, 4, 150	sylancr 589	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
152	147, 149, 151	3eqtr4d 2868	1 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ⟨cop 4575   class class class wbr 5068   × cxp 5555  Rel wrel 5562   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  1^st c1st 7689  2^nd c2nd 7690  ⟨“cs3 14206  Basecbs 16485  Hom chom 16578  compcco 16579  Catccat 16937  Idccid 16938   Func cfunc 17126   FuncCat cfuc 17214   ×_c cxpc 17420   curry_F ccurf 17462   uncurry_F cuncf 17463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-fzo 13037  df-hash 13694  df-word 13865  df-concat 13925  df-s1 13952  df-s2 14212  df-s3 14213  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-hom 16591  df-cco 16592  df-cat 16941  df-cid 16942  df-func 17130  df-cofu 17132  df-nat 17215  df-fuc 17216  df-xpc 17424  df-1stf 17425  df-2ndf 17426  df-prf 17427  df-evlf 17465  df-curf 17466  df-uncf 17467

This theorem is referenced by: (None)