Theorem uncfcurf 18200

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 2729	. . . . . . 7 ⊢ (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
2		uncfcurf.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
3	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
4		uncfcurf.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
5		funcrcl 17825	. . . . . . . . . 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 495	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
8	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐸 ∈ Cat)
9		uncfcurf.g	. . . . . . . . 9 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
10		eqid 2729	. . . . . . . . 9 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
11		uncfcurf.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	9, 10, 11, 2, 4	curfcl 18193	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
13	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
14		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
15		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶))
17		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
18	1, 3, 8, 13, 14, 15, 16, 17	uncf1 18197	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦))
19	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐶 ∈ Cat)
20	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
21		eqid 2729	. . . . . . 7 ⊢ ((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑥)
22	9, 14, 19, 3, 20, 15, 16, 21, 17	curf11 18187	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦) = (𝑥(1st ‘𝐹)𝑦))
23	18, 22	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦))
24	23	ralrimivva 3180	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦))
25		eqid 2729	. . . . . . . 8 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
26	25, 14, 15	xpcbas 18139	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×c 𝐷))
27		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
28		relfunc 17824	. . . . . . . 8 ⊢ Rel ((𝐶 ×c 𝐷) Func 𝐸)
29	1, 2, 7, 12	uncfcl 18196	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) ∈ ((𝐶 ×c 𝐷) Func 𝐸))
30		1st2ndbr 8021	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)))
32	26, 27, 31	funcf1 17828	. . . . . 6 ⊢ (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸))
33	32	ffnd 6689	. . . . 5 ⊢ (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷)))
34		1st2ndbr 8021	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹))
36	26, 27, 35	funcf1 17828	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸))
37	36	ffnd 6689	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐷)))
38		eqfnov2 7519	. . . . 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 584	. . . 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 257	. . 3 ⊢ (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (1st ‘𝐹))
41	2	ad3antrrr 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 730	. . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶))
45	44	adantr 480	. . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
47	46	adantr 480	. . . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
49		eqid 2729	. . . . . . . . . . 11 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
50		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐶))
51	50	adantr 480	. . . . . . . . . . 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 772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑤 ∈ (Base‘𝐷))
53	52	adantr 480	. . . . . . . . . . 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 770	. . . . . . . . . . 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 772	. . . . . . . . . . 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 18198	. . . . . . . . . 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 730	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . 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 18187	. . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . 15 ⊢ (𝑥(1st ‘𝐹)𝑦) = ((1st ‘𝐹)‘⟨𝑥, 𝑦⟩)
61	59, 60	eqtrdi 2780	. . . . . . . . . . . . . 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 18187	. . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . 15 ⊢ (𝑥(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘⟨𝑥, 𝑤⟩)
64	62, 63	eqtrdi 2780	. . . . . . . . . . . . . 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 4845	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝐺)‘𝑧) = ((1st ‘𝐺)‘𝑧)
67	9, 14, 57, 41, 58, 15, 51, 66, 53	curf11 18187	. . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . 14 ⊢ (𝑧(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘⟨𝑧, 𝑤⟩)
69	67, 68	eqtrdi 2780	. . . . . . . . . . . . 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 7405	. . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ (Id‘𝐷) = (Id‘𝐷)
72		eqid 2729	. . . . . . . . . . . . . 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 18191	. . . . . . . . . . . . 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 7390	. . . . . . . . . . . . 13 ⊢ (𝑓(⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑥, 𝑤⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)
75	73, 74	eqtrdi 2780	. . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ (Id‘𝐶) = (Id‘𝐶)
77	9, 14, 57, 41, 58, 15, 45, 21, 47, 49, 76, 53, 55	curf12 18188	. . . . . . . . . . . . 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 7390	. . . . . . . . . . . . 13 ⊢ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑤⟩)𝑔) = ((⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑔⟩)
79	77, 78	eqtrdi 2780	. . . . . . . . . . . 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 7408	. . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
82		eqid 2729	. . . . . . . . . . . 12 ⊢ (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
83		eqid 2729	. . . . . . . . . . . 12 ⊢ (comp‘𝐸) = (comp‘𝐸)
84	35	ad2antrr 726	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 5675	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
87	86	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
88	87	adantr 480	. . . . . . . . . . . 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 5677	. . . . . . . . . . . 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 5675	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
91	90	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
92	91	adantr 480	. . . . . . . . . . . 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 17643	. . . . . . . . . . . . . 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 5677	. . . . . . . . . . . . 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 18147	. . . . . . . . . . . . 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 2831	. . . . . . . . . . . 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 17643	. . . . . . . . . . . . . 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 5677	. . . . . . . . . . . . 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 18147	. . . . . . . . . . . . 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 2831	. . . . . . . . . . . 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 17833	. . . . . . . . . . 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 2729	. . . . . . . . . . . . . . 15 ⊢ (comp‘𝐶) = (comp‘𝐶)
103		eqid 2729	. . . . . . . . . . . . . . 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 18148	. . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . 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 7390	. . . . . . . . . . . . 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	eqtr4di 2782	. . . . . . . . . . . 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 17645	. . . . . . . . . . . . 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 17644	. . . . . . . . . . . . 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 7405	. . . . . . . . . . . 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 2764	. . . . . . . . . . 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 2770	. . . . . . . . . 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 2764	. . . . . . . . 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 3180	. . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
116	31	ad2antrr 726	. . . . . . . . . . . 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 17830	. . . . . . . . . . 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 18147	. . . . . . . . . . . 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 6672	. . . . . . . . . . 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 232	. . . . . . . . . 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 6689	. . . . . . . . 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 17830	. . . . . . . . . . 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 6672	. . . . . . . . . . 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 232	. . . . . . . . . 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 6689	. . . . . . . . 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 7519	. . . . . . . . 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 584	. . . . . . . 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 257	. . . . . . 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 3180	. . . . . 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 3180	. . . . 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 7395	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩))
132		oveq2 7395	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑢(2nd ‘𝐹)𝑣) = (𝑢(2nd ‘𝐹)⟨𝑧, 𝑤⟩))
133	131, 132	eqeq12d 2745	. . . . . . . 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 5805	. . . . . . 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 7394	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩))
136		oveq1 7394	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢(2nd ‘𝐹)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑧, 𝑤⟩))
137	135, 136	eqeq12d 2745	. . . . . . . 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	bitrid 283	. . . . . 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 5805	. . . . 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 234	. . . 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 17831	. . . . 5 ⊢ (𝜑 → (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
143	26, 35	funcfn2 17831	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
144		eqfnov2 7519	. . . . 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 584	. . . 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 257	. . 3 ⊢ (𝜑 → (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (2nd ‘𝐹))
147	40, 146	opeq12d 4845	. 2 ⊢ (𝜑 → ⟨(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)), (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟩ = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
148		1st2nd 8018	. . 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 587	. 2 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = ⟨(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)), (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟩)
150		1st2nd 8018	. . 3 ⊢ ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
151	28, 4, 150	sylancr 587	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
152	147, 149, 151	3eqtr4d 2774	1 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ⟨cop 4595   class class class wbr 5107   × cxp 5636  Rel wrel 5643   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  ⟨“cs3 14808  Basecbs 17179  Hom chom 17231  compcco 17232  Catccat 17625  Idccid 17626   Func cfunc 17816   FuncCat cfuc 17907   ×_c cxpc 18129   curry_F ccurf 18171   uncurry_F cuncf 18172

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-fzo 13616  df-hash 14296  df-word 14479  df-concat 14536  df-s1 14561  df-s2 14814  df-s3 14815  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17629  df-cid 17630  df-func 17820  df-cofu 17822  df-nat 17908  df-fuc 17909  df-xpc 18133  df-1stf 18134  df-2ndf 18135  df-prf 18136  df-evlf 18174  df-curf 18175  df-uncf 18176

This theorem is referenced by: (None)