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Theorem uncfcurf 16702
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 (𝜑𝐶 ∈ Cat)
uncfcurf.d (𝜑𝐷 ∈ Cat)
uncfcurf.f (𝜑𝐹 ∈ ((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.
StepHypRef Expression
1 eqid 2610 . . . . . . 7 (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
2 uncfcurf.d . . . . . . . 8 (𝜑𝐷 ∈ Cat)
32adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
4 uncfcurf.f . . . . . . . . . 10 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
5 funcrcl 16346 . . . . . . . . . 10 (𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸) → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
64, 5syl 17 . . . . . . . . 9 (𝜑 → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
76simprd 478 . . . . . . . 8 (𝜑𝐸 ∈ Cat)
87adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐸 ∈ Cat)
9 uncfcurf.g . . . . . . . . 9 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
10 eqid 2610 . . . . . . . . 9 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
11 uncfcurf.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
129, 10, 11, 2, 4curfcl 16695 . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
1312adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
14 eqid 2610 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
15 eqid 2610 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
16 simprl 790 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶))
17 simprr 792 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
181, 3, 8, 13, 14, 15, 16, 17uncf1 16699 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = ((1st ‘((1st𝐺)‘𝑥))‘𝑦))
1911adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐶 ∈ Cat)
204adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
21 eqid 2610 . . . . . . 7 ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑥)
229, 14, 19, 3, 20, 15, 16, 21, 17curf11 16689 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘((1st𝐺)‘𝑥))‘𝑦) = (𝑥(1st𝐹)𝑦))
2318, 22eqtrd 2644 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st𝐹)𝑦))
2423ralrimivva 2954 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st𝐹)𝑦))
25 eqid 2610 . . . . . . . 8 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
2625, 14, 15xpcbas 16641 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×c 𝐷))
27 eqid 2610 . . . . . . 7 (Base‘𝐸) = (Base‘𝐸)
28 relfunc 16345 . . . . . . . 8 Rel ((𝐶 ×c 𝐷) Func 𝐸)
291, 2, 7, 12uncfcl 16698 . . . . . . . 8 (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) ∈ ((𝐶 ×c 𝐷) Func 𝐸))
30 1st2ndbr 7108 . . . . . . . 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)))
3128, 29, 30sylancr 694 . . . . . . 7 (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)))
3226, 27, 31funcf1 16349 . . . . . 6 (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸))
33 ffn 5958 . . . . . 6 ((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸) → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷)))
3432, 33syl 17 . . . . 5 (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷)))
35 1st2ndbr 7108 . . . . . . . 8 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
3628, 4, 35sylancr 694 . . . . . . 7 (𝜑 → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
3726, 27, 36funcf1 16349 . . . . . 6 (𝜑 → (1st𝐹):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸))
38 ffn 5958 . . . . . 6 ((1st𝐹):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸) → (1st𝐹) Fn ((Base‘𝐶) × (Base‘𝐷)))
3937, 38syl 17 . . . . 5 (𝜑 → (1st𝐹) Fn ((Base‘𝐶) × (Base‘𝐷)))
40 eqfnov2 6665 . . . . 5 (((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷)) ∧ (1st𝐹) Fn ((Base‘𝐶) × (Base‘𝐷))) → ((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (1st𝐹) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st𝐹)𝑦)))
4134, 39, 40syl2anc 691 . . . 4 (𝜑 → ((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (1st𝐹) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st𝐹)𝑦)))
4224, 41mpbird 246 . . 3 (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (1st𝐹))
432ad3antrrr 762 . . . . . . . . . . 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)
447ad3antrrr 762 . . . . . . . . . . 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)
4512ad3antrrr 762 . . . . . . . . . . 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)))
4616adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶))
4746adantr 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))
4817adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
4948adantr 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))
50 eqid 2610 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
51 eqid 2610 . . . . . . . . . . 11 (Hom ‘𝐷) = (Hom ‘𝐷)
52 simprl 790 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐶))
5352adantr 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))
54 simprr 792 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑤 ∈ (Base‘𝐷))
5554adantr 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))
56 simprl 790 . . . . . . . . . . 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))
57 simprr 792 . . . . . . . . . . 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))
581, 43, 44, 45, 14, 15, 47, 49, 50, 51, 53, 55, 56, 57uncf2 16700 . . . . . . . . . 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(2nd𝐺)𝑧)‘𝑓)‘𝑤)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑦), ((1st ‘((1st𝐺)‘𝑥))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑧))‘𝑤))((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑤)‘𝑔)))
5911ad3antrrr 762 . . . . . . . . . . . . . . . 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)
604ad3antrrr 762 . . . . . . . . . . . . . . . 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))
619, 14, 59, 43, 60, 15, 47, 21, 49curf11 16689 . . . . . . . . . . . . . . 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 ‘((1st𝐺)‘𝑥))‘𝑦) = (𝑥(1st𝐹)𝑦))
62 df-ov 6552 . . . . . . . . . . . . . . 15 (𝑥(1st𝐹)𝑦) = ((1st𝐹)‘⟨𝑥, 𝑦⟩)
6361, 62syl6eq 2660 . . . . . . . . . . . . . 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 ‘((1st𝐺)‘𝑥))‘𝑦) = ((1st𝐹)‘⟨𝑥, 𝑦⟩))
649, 14, 59, 43, 60, 15, 47, 21, 55curf11 16689 . . . . . . . . . . . . . . 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 ‘((1st𝐺)‘𝑥))‘𝑤) = (𝑥(1st𝐹)𝑤))
65 df-ov 6552 . . . . . . . . . . . . . . 15 (𝑥(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑥, 𝑤⟩)
6664, 65syl6eq 2660 . . . . . . . . . . . . . 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 ‘((1st𝐺)‘𝑥))‘𝑤) = ((1st𝐹)‘⟨𝑥, 𝑤⟩))
6763, 66opeq12d 4348 . . . . . . . . . . . . 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 ‘((1st𝐺)‘𝑥))‘𝑦), ((1st ‘((1st𝐺)‘𝑥))‘𝑤)⟩ = ⟨((1st𝐹)‘⟨𝑥, 𝑦⟩), ((1st𝐹)‘⟨𝑥, 𝑤⟩)⟩)
68 eqid 2610 . . . . . . . . . . . . . . 15 ((1st𝐺)‘𝑧) = ((1st𝐺)‘𝑧)
699, 14, 59, 43, 60, 15, 53, 68, 55curf11 16689 . . . . . . . . . . . . . 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 ‘((1st𝐺)‘𝑧))‘𝑤) = (𝑧(1st𝐹)𝑤))
70 df-ov 6552 . . . . . . . . . . . . . 14 (𝑧(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑧, 𝑤⟩)
7169, 70syl6eq 2660 . . . . . . . . . . . . 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 ‘((1st𝐺)‘𝑧))‘𝑤) = ((1st𝐹)‘⟨𝑧, 𝑤⟩))
7267, 71oveq12d 6567 . . . . . . . . . . . 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 ‘((1st𝐺)‘𝑥))‘𝑦), ((1st ‘((1st𝐺)‘𝑥))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑧))‘𝑤)) = (⟨((1st𝐹)‘⟨𝑥, 𝑦⟩), ((1st𝐹)‘⟨𝑥, 𝑤⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑧, 𝑤⟩)))
73 eqid 2610 . . . . . . . . . . . . . 14 (Id‘𝐷) = (Id‘𝐷)
74 eqid 2610 . . . . . . . . . . . . . 14 ((𝑥(2nd𝐺)𝑧)‘𝑓) = ((𝑥(2nd𝐺)𝑧)‘𝑓)
759, 14, 59, 43, 60, 15, 50, 73, 47, 53, 56, 74, 55curf2val 16693 . . . . . . . . . . . . 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(2nd𝐺)𝑧)‘𝑓)‘𝑤) = (𝑓(⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)))
76 df-ov 6552 . . . . . . . . . . . . 13 (𝑓(⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)
7775, 76syl6eq 2660 . . . . . . . . . . . 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(2nd𝐺)𝑧)‘𝑓)‘𝑤) = ((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩))
78 eqid 2610 . . . . . . . . . . . . . 14 (Id‘𝐶) = (Id‘𝐶)
799, 14, 59, 43, 60, 15, 47, 21, 49, 51, 78, 55, 57curf12 16690 . . . . . . . . . . . . 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 ‘((1st𝐺)‘𝑥))𝑤)‘𝑔) = (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑤⟩)𝑔))
80 df-ov 6552 . . . . . . . . . . . . 13 (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑤⟩)𝑔) = ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑔⟩)
8179, 80syl6eq 2660 . . . . . . . . . . . 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 ‘((1st𝐺)‘𝑥))𝑤)‘𝑔) = ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑔⟩))
8272, 77, 81oveq123d 6570 . . . . . . . . . . 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(2nd𝐺)𝑧)‘𝑓)‘𝑤)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑦), ((1st ‘((1st𝐺)‘𝑥))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑧))‘𝑤))((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑤)‘𝑔)) = (((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)(⟨((1st𝐹)‘⟨𝑥, 𝑦⟩), ((1st𝐹)‘⟨𝑥, 𝑤⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑧, 𝑤⟩))((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑔⟩)))
83 eqid 2610 . . . . . . . . . . . 12 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
84 eqid 2610 . . . . . . . . . . . 12 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
85 eqid 2610 . . . . . . . . . . . 12 (comp‘𝐸) = (comp‘𝐸)
8636ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
8786adantr 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))) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
88 opelxpi 5072 . . . . . . . . . . . . . 14 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
8988ad2antlr 759 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
9089adantr 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)))
91 opelxpi 5072 . . . . . . . . . . . . 13 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
9247, 55, 91syl2anc 691 . . . . . . . . . . . 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)))
93 opelxpi 5072 . . . . . . . . . . . . . 14 ((𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
9493adantl 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
9594adantr 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)))
9614, 50, 78, 59, 47catidcl 16166 . . . . . . . . . . . . . 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))
97 opelxpi 5072 . . . . . . . . . . . . . 14 ((((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤)) → ⟨((Id‘𝐶)‘𝑥), 𝑔⟩ ∈ ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑤)))
9896, 57, 97syl2anc 691 . . . . . . . . . . . . 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)))
9925, 14, 15, 50, 51, 47, 49, 47, 55, 83xpchom2 16649 . . . . . . . . . . . . 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)))
10098, 99eleqtrrd 2691 . . . . . . . . . . . 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⟩))
10115, 51, 73, 43, 55catidcl 16166 . . . . . . . . . . . . . 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))
102 opelxpi 5072 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ((Id‘𝐷)‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤)) → ⟨𝑓, ((Id‘𝐷)‘𝑤)⟩ ∈ ((𝑥(Hom ‘𝐶)𝑧) × (𝑤(Hom ‘𝐷)𝑤)))
10356, 101, 102syl2anc 691 . . . . . . . . . . . . 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)))
10425, 14, 15, 50, 51, 47, 55, 53, 55, 83xpchom2 16649 . . . . . . . . . . . . 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)))
105103, 104eleqtrrd 2691 . . . . . . . . . . . 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⟩))
10626, 83, 84, 85, 87, 90, 92, 95, 100, 105funcco 16354 . . . . . . . . . . 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⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑓, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑦⟩, ⟨𝑥, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨((Id‘𝐶)‘𝑥), 𝑔⟩)) = (((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)(⟨((1st𝐹)‘⟨𝑥, 𝑦⟩), ((1st𝐹)‘⟨𝑥, 𝑤⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑧, 𝑤⟩))((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑔⟩)))
107 eqid 2610 . . . . . . . . . . . . . . 15 (comp‘𝐶) = (comp‘𝐶)
108 eqid 2610 . . . . . . . . . . . . . . 15 (comp‘𝐷) = (comp‘𝐷)
10925, 14, 15, 50, 51, 47, 49, 47, 55, 107, 108, 84, 53, 55, 96, 57, 56, 101xpcco2 16650 . . . . . . . . . . . . . 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)⟩)
110109fveq2d 6107 . . . . . . . . . . . . 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⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑓, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑦⟩, ⟨𝑥, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨((Id‘𝐶)‘𝑥), 𝑔⟩)) = ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥)), (((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔)⟩))
111 df-ov 6552 . . . . . . . . . . . . 13 ((𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥))(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩)(((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔)) = ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥)), (((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔)⟩)
112110, 111syl6eqr 2662 . . . . . . . . . . . 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⟩(2nd𝐹)⟨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⟩(2nd𝐹)⟨𝑧, 𝑤⟩)(((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔)))
11314, 50, 78, 59, 47, 107, 53, 56catrid 16168 . . . . . . . . . . . . 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)
11415, 51, 73, 43, 49, 108, 55, 57catlid 16167 . . . . . . . . . . . . 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)
115113, 114oveq12d 6567 . . . . . . . . . . . 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⟩(2nd𝐹)⟨𝑧, 𝑤⟩)(((Id‘𝐷)‘𝑤)(⟨𝑦, 𝑤⟩(comp‘𝐷)𝑤)𝑔)) = (𝑓(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩)𝑔))
116112, 115eqtrd 2644 . . . . . . . . . . 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⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑓, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑦⟩, ⟨𝑥, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨((Id‘𝐶)‘𝑥), 𝑔⟩)) = (𝑓(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩)𝑔))
11782, 106, 1163eqtr2d 2650 . . . . . . . . . 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(2nd𝐺)𝑧)‘𝑓)‘𝑤)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑦), ((1st ‘((1st𝐺)‘𝑥))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑧))‘𝑤))((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑤)‘𝑔)) = (𝑓(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩)𝑔))
11858, 117eqtrd 2644 . . . . . . . . 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⟩(2nd𝐹)⟨𝑧, 𝑤⟩)𝑔))
119118ralrimivva 2954 . . . . . . . 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⟩(2nd𝐹)⟨𝑧, 𝑤⟩)𝑔))
120 eqid 2610 . . . . . . . . . . . 12 (Hom ‘𝐸) = (Hom ‘𝐸)
12131ad2antrr 758 . . . . . . . . . . . 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)))
12226, 83, 120, 121, 89, 94funcf2 16351 . . . . . . . . . . 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⟩)))
12325, 14, 15, 50, 51, 46, 48, 52, 54, 83xpchom2 16649 . . . . . . . . . . . 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)))
124123feq2d 5944 . . . . . . . . . . 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⟩))))
125122, 124mpbid 221 . . . . . . . . . 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⟩)))
126 ffn 5958 . . . . . . . . . 10 ((⟨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⟩)) → (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤)))
127125, 126syl 17 . . . . . . . . 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)))
12826, 83, 120, 86, 89, 94funcf2 16351 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩):(⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟶(((1st𝐹)‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑧, 𝑤⟩)))
129123feq2d 5944 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩):(⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟶(((1st𝐹)‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑧, 𝑤⟩)) ↔ (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st𝐹)‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑧, 𝑤⟩))))
130128, 129mpbid 221 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st𝐹)‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑧, 𝑤⟩)))
131 ffn 5958 . . . . . . . . . 10 ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st𝐹)‘⟨𝑥, 𝑦⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑧, 𝑤⟩)) → (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤)))
132130, 131syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤)))
133 eqfnov2 6665 . . . . . . . . 9 (((⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤)) ∧ (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))) → ((⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨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⟩(2nd𝐹)⟨𝑧, 𝑤⟩)𝑔)))
134127, 132, 133syl2anc 691 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ((⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨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⟩(2nd𝐹)⟨𝑧, 𝑤⟩)𝑔)))
135119, 134mpbird 246 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩))
136135ralrimivva 2954 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩))
137136ralrimivva 2954 . . . . 5 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩))
138 oveq2 6557 . . . . . . . . 9 (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩))
139 oveq2 6557 . . . . . . . . 9 (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑢(2nd𝐹)𝑣) = (𝑢(2nd𝐹)⟨𝑧, 𝑤⟩))
140138, 139eqeq12d 2625 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd𝐹)𝑣) ↔ (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (𝑢(2nd𝐹)⟨𝑧, 𝑤⟩)))
141140ralxp 5185 . . . . . . 7 (∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd𝐹)𝑣) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (𝑢(2nd𝐹)⟨𝑧, 𝑤⟩))
142 oveq1 6556 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩))
143 oveq1 6556 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢(2nd𝐹)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩))
144142, 143eqeq12d 2625 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (𝑢(2nd𝐹)⟨𝑧, 𝑤⟩) ↔ (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩)))
1451442ralbidv 2972 . . . . . . 7 (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (𝑢(2nd𝐹)⟨𝑧, 𝑤⟩) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩)))
146141, 145syl5bb 271 . . . . . 6 (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd𝐹)𝑣) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩)))
147146ralxp 5185 . . . . 5 (∀𝑢 ∈ ((Base‘𝐶) × (Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd𝐹)𝑣) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩))
148137, 147sylibr 223 . . . 4 (𝜑 → ∀𝑢 ∈ ((Base‘𝐶) × (Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd𝐹)𝑣))
14926, 31funcfn2 16352 . . . . 5 (𝜑 → (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
15026, 36funcfn2 16352 . . . . 5 (𝜑 → (2nd𝐹) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
151 eqfnov2 6665 . . . . 5 (((2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))) ∧ (2nd𝐹) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))) → ((2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (2nd𝐹) ↔ ∀𝑢 ∈ ((Base‘𝐶) × (Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd𝐹)𝑣)))
152149, 150, 151syl2anc 691 . . . 4 (𝜑 → ((2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (2nd𝐹) ↔ ∀𝑢 ∈ ((Base‘𝐶) × (Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd𝐹)𝑣)))
153148, 152mpbird 246 . . 3 (𝜑 → (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (2nd𝐹))
15442, 153opeq12d 4348 . 2 (𝜑 → ⟨(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)), (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟩ = ⟨(1st𝐹), (2nd𝐹)⟩)
155 1st2nd 7105 . . 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))⟩)
15628, 29, 155sylancr 694 . 2 (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = ⟨(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)), (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟩)
157 1st2nd 7105 . . 3 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
15828, 4, 157sylancr 694 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
159154, 156, 1583eqtr4d 2654 1 (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  cop 4131   class class class wbr 4583   × cxp 5036  Rel wrel 5043   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  ⟨“cs3 13438  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149   Func cfunc 16337   FuncCat cfuc 16425   ×c cxpc 16631   curryF ccurf 16673   uncurryF cuncf 16674
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-card 8648  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-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  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-func 16341  df-cofu 16343  df-nat 16426  df-fuc 16427  df-xpc 16635  df-1stf 16636  df-2ndf 16637  df-prf 16638  df-evlf 16676  df-curf 16677  df-uncf 16678
This theorem is referenced by: (None)
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