MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uncfcurf Structured version   Visualization version   GIF version

Theorem uncfcurf 18240
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 2728 . . . . . . 7 (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
2 uncfcurf.d . . . . . . . 8 (𝜑𝐷 ∈ Cat)
32adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat)
4 uncfcurf.f . . . . . . . . . 10 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
5 funcrcl 17858 . . . . . . . . . 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 494 . . . . . . . 8 (𝜑𝐸 ∈ Cat)
87adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐸 ∈ Cat)
9 uncfcurf.g . . . . . . . . 9 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
10 eqid 2728 . . . . . . . . 9 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
11 uncfcurf.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
129, 10, 11, 2, 4curfcl 18233 . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
1312adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
14 eqid 2728 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
15 eqid 2728 . . . . . . 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))
181, 3, 8, 13, 14, 15, 16, 17uncf1 18237 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = ((1st ‘((1st𝐺)‘𝑥))‘𝑦))
1911adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐶 ∈ Cat)
204adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
21 eqid 2728 . . . . . . 7 ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑥)
229, 14, 19, 3, 20, 15, 16, 21, 17curf11 18227 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st ‘((1st𝐺)‘𝑥))‘𝑦) = (𝑥(1st𝐹)𝑦))
2318, 22eqtrd 2768 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st𝐹)𝑦))
2423ralrimivva 3198 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st𝐹)𝑦))
25 eqid 2728 . . . . . . . 8 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
2625, 14, 15xpcbas 18178 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×c 𝐷))
27 eqid 2728 . . . . . . 7 (Base‘𝐸) = (Base‘𝐸)
28 relfunc 17857 . . . . . . . 8 Rel ((𝐶 ×c 𝐷) Func 𝐸)
291, 2, 7, 12uncfcl 18236 . . . . . . . 8 (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) ∈ ((𝐶 ×c 𝐷) Func 𝐸))
30 1st2ndbr 8054 . . . . . . . 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 585 . . . . . . 7 (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)))
3226, 27, 31funcf1 17861 . . . . . 6 (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸))
3332ffnd 6728 . . . . 5 (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷)))
34 1st2ndbr 8054 . . . . . . . 8 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
3528, 4, 34sylancr 585 . . . . . . 7 (𝜑 → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
3626, 27, 35funcf1 17861 . . . . . 6 (𝜑 → (1st𝐹):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸))
3736ffnd 6728 . . . . 5 (𝜑 → (1st𝐹) Fn ((Base‘𝐶) × (Base‘𝐷)))
38 eqfnov2 7558 . . . . 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)))
3933, 37, 38syl2anc 582 . . . 4 (𝜑 → ((1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (1st𝐹) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑦) = (𝑥(1st𝐹)𝑦)))
4024, 39mpbird 256 . . 3 (𝜑 → (1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (1st𝐹))
412ad3antrrr 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)
427ad3antrrr 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)
4312ad3antrrr 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)))
4416adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶))
4544adantr 479 . . . . . . . . . . 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))
4617adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
4746adantr 479 . . . . . . . . . . 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 2728 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
49 eqid 2728 . . . . . . . . . . 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))
5150adantr 479 . . . . . . . . . . 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))
5352adantr 479 . . . . . . . . . . 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))
561, 41, 42, 43, 14, 15, 45, 47, 48, 49, 51, 53, 54, 55uncf2 18238 . . . . . . . . . 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)))
5711ad3antrrr 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)
584ad3antrrr 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))
599, 14, 57, 41, 58, 15, 45, 21, 47curf11 18227 . . . . . . . . . . . . . . 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))
60 df-ov 7429 . . . . . . . . . . . . . . 15 (𝑥(1st𝐹)𝑦) = ((1st𝐹)‘⟨𝑥, 𝑦⟩)
6159, 60eqtrdi 2784 . . . . . . . . . . . . . 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⟩))
629, 14, 57, 41, 58, 15, 45, 21, 53curf11 18227 . . . . . . . . . . . . . . 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))
63 df-ov 7429 . . . . . . . . . . . . . . 15 (𝑥(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑥, 𝑤⟩)
6462, 63eqtrdi 2784 . . . . . . . . . . . . . 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⟩))
6561, 64opeq12d 4886 . . . . . . . . . . . . 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⟩)⟩)
66 eqid 2728 . . . . . . . . . . . . . . 15 ((1st𝐺)‘𝑧) = ((1st𝐺)‘𝑧)
679, 14, 57, 41, 58, 15, 51, 66, 53curf11 18227 . . . . . . . . . . . . . 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))
68 df-ov 7429 . . . . . . . . . . . . . 14 (𝑧(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑧, 𝑤⟩)
6967, 68eqtrdi 2784 . . . . . . . . . . . . 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⟩))
7065, 69oveq12d 7444 . . . . . . . . . . . 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⟩)))
71 eqid 2728 . . . . . . . . . . . . . 14 (Id‘𝐷) = (Id‘𝐷)
72 eqid 2728 . . . . . . . . . . . . . 14 ((𝑥(2nd𝐺)𝑧)‘𝑓) = ((𝑥(2nd𝐺)𝑧)‘𝑓)
739, 14, 57, 41, 58, 15, 48, 71, 45, 51, 54, 72, 53curf2val 18231 . . . . . . . . . . . . 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)))
74 df-ov 7429 . . . . . . . . . . . . 13 (𝑓(⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)
7573, 74eqtrdi 2784 . . . . . . . . . . . 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)⟩))
76 eqid 2728 . . . . . . . . . . . . . 14 (Id‘𝐶) = (Id‘𝐶)
779, 14, 57, 41, 58, 15, 45, 21, 47, 49, 76, 53, 55curf12 18228 . . . . . . . . . . . . 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))
78 df-ov 7429 . . . . . . . . . . . . 13 (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑤⟩)𝑔) = ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑔⟩)
7977, 78eqtrdi 2784 . . . . . . . . . . . 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⟩))
8070, 75, 79oveq123d 7447 . . . . . . . . . . 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⟩)))
81 eqid 2728 . . . . . . . . . . . 12 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
82 eqid 2728 . . . . . . . . . . . 12 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
83 eqid 2728 . . . . . . . . . . . 12 (comp‘𝐸) = (comp‘𝐸)
8435ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
8584adantr 479 . . . . . . . . . . . 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))
86 opelxpi 5719 . . . . . . . . . . . . . 14 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
8786ad2antlr 725 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
8887adantr 479 . . . . . . . . . . . 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)))
8945, 53opelxpd 5721 . . . . . . . . . . . 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 5719 . . . . . . . . . . . . . 14 ((𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
9190adantl 480 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
9291adantr 479 . . . . . . . . . . . 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)))
9314, 48, 76, 57, 45catidcl 17671 . . . . . . . . . . . . . 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))
9493, 55opelxpd 5721 . . . . . . . . . . . . 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)))
9525, 14, 15, 48, 49, 45, 47, 45, 53, 81xpchom2 18186 . . . . . . . . . . . . 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)))
9694, 95eleqtrrd 2832 . . . . . . . . . . . 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⟩))
9715, 49, 71, 41, 53catidcl 17671 . . . . . . . . . . . . . 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))
9854, 97opelxpd 5721 . . . . . . . . . . . . 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)))
9925, 14, 15, 48, 49, 45, 53, 51, 53, 81xpchom2 18186 . . . . . . . . . . . . 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)))
10098, 99eleqtrrd 2832 . . . . . . . . . . . 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⟩))
10126, 81, 82, 83, 85, 88, 89, 92, 96, 100funcco 17866 . . . . . . . . . . 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⟩)))
102 eqid 2728 . . . . . . . . . . . . . . 15 (comp‘𝐶) = (comp‘𝐶)
103 eqid 2728 . . . . . . . . . . . . . . 15 (comp‘𝐷) = (comp‘𝐷)
10425, 14, 15, 48, 49, 45, 47, 45, 53, 102, 103, 82, 51, 53, 93, 55, 54, 97xpcco2 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))) → (⟨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)⟩)
105104fveq2d 6906 . . . . . . . . . . . . 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)⟩))
106 df-ov 7429 . . . . . . . . . . . . 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)⟩)
107105, 106eqtr4di 2786 . . . . . . . . . . . 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)))
10814, 48, 76, 57, 45, 102, 51, 54catrid 17673 . . . . . . . . . . . . 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)
10915, 49, 71, 41, 47, 103, 53, 55catlid 17672 . . . . . . . . . . . . 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)
110108, 109oveq12d 7444 . . . . . . . . . . . 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))
111107, 110eqtrd 2768 . . . . . . . . . . 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))
11280, 101, 1113eqtr2d 2774 . . . . . . . . . 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))
11356, 112eqtrd 2768 . . . . . . . . 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))
114113ralrimivva 3198 . . . . . . . 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))
115 eqid 2728 . . . . . . . . . . . 12 (Hom ‘𝐸) = (Hom ‘𝐸)
11631ad2antrr 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)))
11726, 81, 115, 116, 87, 91funcf2 17863 . . . . . . . . . . 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⟩)))
11825, 14, 15, 48, 49, 44, 46, 50, 52, 81xpchom2 18186 . . . . . . . . . . . 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)))
119118feq2d 6713 . . . . . . . . . . 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⟩))))
120117, 119mpbid 231 . . . . . . . . . 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⟩)))
121120ffnd 6728 . . . . . . . . 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)))
12226, 81, 115, 84, 87, 91funcf2 17863 . . . . . . . . . . 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⟩)))
123118feq2d 6713 . . . . . . . . . . 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⟩))))
124122, 123mpbid 231 . . . . . . . . . 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⟩)))
125124ffnd 6728 . . . . . . . . 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)))
126 eqfnov2 7558 . . . . . . . . 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)))
127121, 125, 126syl2anc 582 . . . . . . . 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)))
128114, 127mpbird 256 . . . . . . 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⟩))
129128ralrimivva 3198 . . . . . 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⟩))
130129ralrimivva 3198 . . . . 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⟩))
131 oveq2 7434 . . . . . . . . 9 (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩))
132 oveq2 7434 . . . . . . . . 9 (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑢(2nd𝐹)𝑣) = (𝑢(2nd𝐹)⟨𝑧, 𝑤⟩))
133131, 132eqeq12d 2744 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd𝐹)𝑣) ↔ (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (𝑢(2nd𝐹)⟨𝑧, 𝑤⟩)))
134133ralxp 5848 . . . . . . 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⟩))
135 oveq1 7433 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟨𝑧, 𝑤⟩))
136 oveq1 7433 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢(2nd𝐹)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑧, 𝑤⟩))
137135, 136eqeq12d 2744 . . . . . . . 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⟩)))
1381372ralbidv 3216 . . . . . . 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⟩)))
139134, 138bitrid 282 . . . . . 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⟩)))
140139ralxp 5848 . . . . 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⟩))
141130, 140sylibr 233 . . . 4 (𝜑 → ∀𝑢 ∈ ((Base‘𝐶) × (Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))𝑣) = (𝑢(2nd𝐹)𝑣))
14226, 31funcfn2 17864 . . . . 5 (𝜑 → (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
14326, 35funcfn2 17864 . . . . 5 (𝜑 → (2nd𝐹) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
144 eqfnov2 7558 . . . . 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)))
145142, 143, 144syl2anc 582 . . . 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)))
146141, 145mpbird 256 . . 3 (𝜑 → (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)) = (2nd𝐹))
14740, 146opeq12d 4886 . 2 (𝜑 → ⟨(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)), (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟩ = ⟨(1st𝐹), (2nd𝐹)⟩)
148 1st2nd 8051 . . 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))⟩)
14928, 29, 148sylancr 585 . 2 (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = ⟨(1st ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)), (2nd ‘(⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺))⟩)
150 1st2nd 8051 . . 3 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
15128, 4, 150sylancr 585 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
152147, 149, 1513eqtr4d 2778 1 (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058  cop 4638   class class class wbr 5152   × cxp 5680  Rel wrel 5687   Fn wfn 6548  wf 6549  cfv 6553  (class class class)co 7426  1st c1st 7999  2nd c2nd 8000  ⟨“cs3 14835  Basecbs 17189  Hom chom 17253  compcco 17254  Catccat 17653  Idccid 17654   Func cfunc 17849   FuncCat cfuc 17941   ×c cxpc 18168   curryF ccurf 18211   uncurryF cuncf 18212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-er 8733  df-map 8855  df-ixp 8925  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-card 9972  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12253  df-2 12315  df-3 12316  df-4 12317  df-5 12318  df-6 12319  df-7 12320  df-8 12321  df-9 12322  df-n0 12513  df-z 12599  df-dec 12718  df-uz 12863  df-fz 13527  df-fzo 13670  df-hash 14332  df-word 14507  df-concat 14563  df-s1 14588  df-s2 14841  df-s3 14842  df-struct 17125  df-slot 17160  df-ndx 17172  df-base 17190  df-hom 17266  df-cco 17267  df-cat 17657  df-cid 17658  df-func 17853  df-cofu 17855  df-nat 17942  df-fuc 17943  df-xpc 18172  df-1stf 18173  df-2ndf 18174  df-prf 18175  df-evlf 18214  df-curf 18215  df-uncf 18216
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator