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Theorem curf2cl 16694
 Description: The curry functor at a morphism is a natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
Hypotheses
Ref Expression
curf2.g 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curf2.a 𝐴 = (Base‘𝐶)
curf2.c (𝜑𝐶 ∈ Cat)
curf2.d (𝜑𝐷 ∈ Cat)
curf2.f (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curf2.b 𝐵 = (Base‘𝐷)
curf2.h 𝐻 = (Hom ‘𝐶)
curf2.i 𝐼 = (Id‘𝐷)
curf2.x (𝜑𝑋𝐴)
curf2.y (𝜑𝑌𝐴)
curf2.k (𝜑𝐾 ∈ (𝑋𝐻𝑌))
curf2.l 𝐿 = ((𝑋(2nd𝐺)𝑌)‘𝐾)
curf2.n 𝑁 = (𝐷 Nat 𝐸)
Assertion
Ref Expression
curf2cl (𝜑𝐿 ∈ (((1st𝐺)‘𝑋)𝑁((1st𝐺)‘𝑌)))

Proof of Theorem curf2cl
Dummy variables 𝑧 𝑤 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curf2.g . . . 4 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2 curf2.a . . . 4 𝐴 = (Base‘𝐶)
3 curf2.c . . . 4 (𝜑𝐶 ∈ Cat)
4 curf2.d . . . 4 (𝜑𝐷 ∈ Cat)
5 curf2.f . . . 4 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6 curf2.b . . . 4 𝐵 = (Base‘𝐷)
7 curf2.h . . . 4 𝐻 = (Hom ‘𝐶)
8 curf2.i . . . 4 𝐼 = (Id‘𝐷)
9 curf2.x . . . 4 (𝜑𝑋𝐴)
10 curf2.y . . . 4 (𝜑𝑌𝐴)
11 curf2.k . . . 4 (𝜑𝐾 ∈ (𝑋𝐻𝑌))
12 curf2.l . . . 4 𝐿 = ((𝑋(2nd𝐺)𝑌)‘𝐾)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12curf2 16692 . . 3 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
14 eqid 2610 . . . . . . . . . 10 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
1514, 2, 6xpcbas 16641 . . . . . . . . 9 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
16 eqid 2610 . . . . . . . . 9 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
17 eqid 2610 . . . . . . . . 9 (Hom ‘𝐸) = (Hom ‘𝐸)
18 relfunc 16345 . . . . . . . . . . 11 Rel ((𝐶 ×c 𝐷) Func 𝐸)
19 1st2ndbr 7108 . . . . . . . . . . 11 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
2018, 5, 19sylancr 694 . . . . . . . . . 10 (𝜑 → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
2120adantr 480 . . . . . . . . 9 ((𝜑𝑧𝐵) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
22 opelxpi 5072 . . . . . . . . . 10 ((𝑋𝐴𝑧𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
239, 22sylan 487 . . . . . . . . 9 ((𝜑𝑧𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
24 opelxpi 5072 . . . . . . . . . 10 ((𝑌𝐴𝑧𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
2510, 24sylan 487 . . . . . . . . 9 ((𝜑𝑧𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
2615, 16, 17, 21, 23, 25funcf2 16351 . . . . . . . 8 ((𝜑𝑧𝐵) → (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩):(⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩)⟶(((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)))
27 eqid 2610 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
289adantr 480 . . . . . . . . . 10 ((𝜑𝑧𝐵) → 𝑋𝐴)
29 simpr 476 . . . . . . . . . 10 ((𝜑𝑧𝐵) → 𝑧𝐵)
3010adantr 480 . . . . . . . . . 10 ((𝜑𝑧𝐵) → 𝑌𝐴)
3114, 2, 6, 7, 27, 28, 29, 30, 29, 16xpchom2 16649 . . . . . . . . 9 ((𝜑𝑧𝐵) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
3231feq2d 5944 . . . . . . . 8 ((𝜑𝑧𝐵) → ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩):(⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩)⟶(((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)) ↔ (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩):((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧))⟶(((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩))))
3326, 32mpbid 221 . . . . . . 7 ((𝜑𝑧𝐵) → (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩):((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧))⟶(((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)))
3411adantr 480 . . . . . . 7 ((𝜑𝑧𝐵) → 𝐾 ∈ (𝑋𝐻𝑌))
354adantr 480 . . . . . . . 8 ((𝜑𝑧𝐵) → 𝐷 ∈ Cat)
366, 27, 8, 35, 29catidcl 16166 . . . . . . 7 ((𝜑𝑧𝐵) → (𝐼𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
3733, 34, 36fovrnd 6704 . . . . . 6 ((𝜑𝑧𝐵) → (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) ∈ (((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)))
383adantr 480 . . . . . . . . 9 ((𝜑𝑧𝐵) → 𝐶 ∈ Cat)
395adantr 480 . . . . . . . . 9 ((𝜑𝑧𝐵) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
40 eqid 2610 . . . . . . . . 9 ((1st𝐺)‘𝑋) = ((1st𝐺)‘𝑋)
411, 2, 38, 35, 39, 6, 28, 40, 29curf11 16689 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = (𝑋(1st𝐹)𝑧))
42 df-ov 6552 . . . . . . . 8 (𝑋(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩)
4341, 42syl6eq 2660 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
44 eqid 2610 . . . . . . . . 9 ((1st𝐺)‘𝑌) = ((1st𝐺)‘𝑌)
451, 2, 38, 35, 39, 6, 30, 44, 29curf11 16689 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = (𝑌(1st𝐹)𝑧))
46 df-ov 6552 . . . . . . . 8 (𝑌(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩)
4745, 46syl6eq 2660 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩))
4843, 47oveq12d 6567 . . . . . 6 ((𝜑𝑧𝐵) → (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)) = (((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)))
4937, 48eleqtrrd 2691 . . . . 5 ((𝜑𝑧𝐵) → (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) ∈ (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
5049ralrimiva 2949 . . . 4 (𝜑 → ∀𝑧𝐵 (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) ∈ (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
51 fvex 6113 . . . . . 6 (Base‘𝐷) ∈ V
526, 51eqeltri 2684 . . . . 5 𝐵 ∈ V
53 mptelixpg 7831 . . . . 5 (𝐵 ∈ V → ((𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))) ∈ X𝑧𝐵 (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)) ↔ ∀𝑧𝐵 (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) ∈ (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧))))
5452, 53ax-mp 5 . . . 4 ((𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))) ∈ X𝑧𝐵 (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)) ↔ ∀𝑧𝐵 (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) ∈ (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
5550, 54sylibr 223 . . 3 (𝜑 → (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))) ∈ X𝑧𝐵 (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
5613, 55eqeltrd 2688 . 2 (𝜑𝐿X𝑧𝐵 (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
57 eqid 2610 . . . . . . . . . 10 (Id‘𝐶) = (Id‘𝐶)
583adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat)
599adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋𝐴)
60 eqid 2610 . . . . . . . . . 10 (comp‘𝐶) = (comp‘𝐶)
6110adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑌𝐴)
6211adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐾 ∈ (𝑋𝐻𝑌))
632, 7, 57, 58, 59, 60, 61, 62catrid 16168 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐾(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐾)
642, 7, 57, 58, 59, 60, 61, 62catlid 16167 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾) = 𝐾)
6563, 64eqtr4d 2647 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐾(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾))
664adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat)
67 simpr1 1060 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧𝐵)
68 eqid 2610 . . . . . . . . . 10 (comp‘𝐷) = (comp‘𝐷)
69 simpr2 1061 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤𝐵)
70 simpr3 1062 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))
716, 27, 8, 66, 67, 68, 69, 70catlid 16167 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓) = 𝑓)
726, 27, 8, 66, 67, 68, 69, 70catrid 16168 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼𝑧)) = 𝑓)
7371, 72eqtr4d 2647 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓) = (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼𝑧)))
7465, 73opeq12d 4348 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨(𝐾(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)), ((𝐼𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓)⟩ = ⟨(((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾), (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼𝑧))⟩)
75 eqid 2610 . . . . . . . 8 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
762, 7, 57, 58, 59catidcl 16166 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
776, 27, 8, 66, 69catidcl 16166 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤))
7814, 2, 6, 7, 27, 59, 67, 59, 69, 60, 68, 75, 61, 69, 76, 70, 62, 77xpcco2 16650 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝐾, (𝐼𝑤)⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑋, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑓⟩) = ⟨(𝐾(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)), ((𝐼𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓)⟩)
79363ad2antr1 1219 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
802, 7, 57, 58, 61catidcl 16166 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐻𝑌))
8114, 2, 6, 7, 27, 59, 67, 61, 67, 60, 68, 75, 61, 69, 62, 79, 80, 70xpcco2 16650 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((Id‘𝐶)‘𝑌), 𝑓⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑌, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩)⟨𝐾, (𝐼𝑧)⟩) = ⟨(((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾), (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼𝑧))⟩)
8274, 78, 813eqtr4d 2654 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝐾, (𝐼𝑤)⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑋, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑓⟩) = (⟨((Id‘𝐶)‘𝑌), 𝑓⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑌, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩)⟨𝐾, (𝐼𝑧)⟩))
8382fveq2d 6107 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘(⟨𝐾, (𝐼𝑤)⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑋, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑓⟩)) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘(⟨((Id‘𝐶)‘𝑌), 𝑓⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑌, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩)⟨𝐾, (𝐼𝑧)⟩)))
84 eqid 2610 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
8520adantr 480 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
86233ad2antr1 1219 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
87 opelxpi 5072 . . . . . . 7 ((𝑋𝐴𝑤𝐵) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
8859, 69, 87syl2anc 691 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
89 opelxpi 5072 . . . . . . 7 ((𝑌𝐴𝑤𝐵) → ⟨𝑌, 𝑤⟩ ∈ (𝐴 × 𝐵))
9061, 69, 89syl2anc 691 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑤⟩ ∈ (𝐴 × 𝐵))
91 opelxpi 5072 . . . . . . . 8 ((((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
9276, 70, 91syl2anc 691 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
9314, 2, 6, 7, 27, 59, 67, 59, 69, 16xpchom2 16649 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩) = ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
9492, 93eleqtrrd 2691 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩))
95 opelxpi 5072 . . . . . . . 8 ((𝐾 ∈ (𝑋𝐻𝑌) ∧ (𝐼𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤)) → ⟨𝐾, (𝐼𝑤)⟩ ∈ ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
9662, 77, 95syl2anc 691 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑤)⟩ ∈ ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
9714, 2, 6, 7, 27, 59, 69, 61, 69, 16xpchom2 16649 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
9896, 97eleqtrrd 2691 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑤)⟩ ∈ (⟨𝑋, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩))
9915, 16, 75, 84, 85, 86, 88, 90, 94, 98funcco 16354 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘(⟨𝐾, (𝐼𝑤)⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑋, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑓⟩)) = (((⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼𝑤)⟩)(⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑋, 𝑤⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)))
100253ad2antr1 1219 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
101 opelxpi 5072 . . . . . . . 8 ((𝐾 ∈ (𝑋𝐻𝑌) ∧ (𝐼𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧)) → ⟨𝐾, (𝐼𝑧)⟩ ∈ ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
10262, 79, 101syl2anc 691 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑧)⟩ ∈ ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
10314, 2, 6, 7, 27, 59, 67, 61, 67, 16xpchom2 16649 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
104102, 103eleqtrrd 2691 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑧)⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩))
105 opelxpi 5072 . . . . . . . 8 ((((Id‘𝐶)‘𝑌) ∈ (𝑌𝐻𝑌) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
10680, 70, 105syl2anc 691 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
10714, 2, 6, 7, 27, 61, 67, 61, 69, 16xpchom2 16649 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑌, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
108106, 107eleqtrrd 2691 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ (⟨𝑌, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩))
10915, 16, 75, 84, 85, 86, 100, 90, 104, 108funcco 16354 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘(⟨((Id‘𝐶)‘𝑌), 𝑓⟩(⟨⟨𝑋, 𝑧⟩, ⟨𝑌, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩)⟨𝐾, (𝐼𝑧)⟩)) = (((⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)(⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑌, 𝑧⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼𝑧)⟩)))
11083, 99, 1093eqtr3d 2652 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼𝑤)⟩)(⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑋, 𝑤⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)) = (((⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)(⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑌, 𝑧⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼𝑧)⟩)))
1115adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
1121, 2, 58, 66, 111, 6, 59, 40, 67curf11 16689 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = (𝑋(1st𝐹)𝑧))
113112, 42syl6eq 2660 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
1141, 2, 58, 66, 111, 6, 59, 40, 69curf11 16689 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑤) = (𝑋(1st𝐹)𝑤))
115 df-ov 6552 . . . . . . . 8 (𝑋(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩)
116114, 115syl6eq 2660 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩))
117113, 116opeq12d 4348 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑋))‘𝑤)⟩ = ⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑋, 𝑤⟩)⟩)
1181, 2, 58, 66, 111, 6, 61, 44, 69curf11 16689 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑤) = (𝑌(1st𝐹)𝑤))
119 df-ov 6552 . . . . . . 7 (𝑌(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑌, 𝑤⟩)
120118, 119syl6eq 2660 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑤) = ((1st𝐹)‘⟨𝑌, 𝑤⟩))
121117, 120oveq12d 6567 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑋))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑤)) = (⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑋, 𝑤⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑌, 𝑤⟩)))
1221, 2, 58, 66, 111, 6, 7, 8, 59, 61, 62, 12, 69curf2val 16693 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑤) = (𝐾(⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)(𝐼𝑤)))
123 df-ov 6552 . . . . . 6 (𝐾(⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)(𝐼𝑤)) = ((⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼𝑤)⟩)
124122, 123syl6eq 2660 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑤) = ((⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼𝑤)⟩))
1251, 2, 58, 66, 111, 6, 59, 40, 67, 27, 57, 69, 70curf12 16690 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑋))𝑤)‘𝑓) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)𝑓))
126 df-ov 6552 . . . . . 6 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)𝑓) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)
127125, 126syl6eq 2660 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑋))𝑤)‘𝑓) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩))
128121, 124, 127oveq123d 6570 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐿𝑤)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑋))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑤))((𝑧(2nd ‘((1st𝐺)‘𝑋))𝑤)‘𝑓)) = (((⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼𝑤)⟩)(⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑋, 𝑤⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)))
1291, 2, 58, 66, 111, 6, 61, 44, 67curf11 16689 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = (𝑌(1st𝐹)𝑧))
130129, 46syl6eq 2660 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩))
131113, 130opeq12d 4348 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑌))‘𝑧)⟩ = ⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑌, 𝑧⟩)⟩)
132131, 120oveq12d 6567 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑌))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑤)) = (⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑌, 𝑧⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑌, 𝑤⟩)))
1331, 2, 58, 66, 111, 6, 61, 44, 67, 27, 57, 69, 70curf12 16690 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑌))𝑤)‘𝑓) = (((Id‘𝐶)‘𝑌)(⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)𝑓))
134 df-ov 6552 . . . . . 6 (((Id‘𝐶)‘𝑌)(⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)𝑓) = ((⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)
135133, 134syl6eq 2660 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑌))𝑤)‘𝑓) = ((⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩))
1361, 2, 58, 66, 111, 6, 7, 8, 59, 61, 62, 12, 67curf2val 16693 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑧) = (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))
137 df-ov 6552 . . . . . 6 (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼𝑧)⟩)
138136, 137syl6eq 2660 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑧) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼𝑧)⟩))
139132, 135, 138oveq123d 6570 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((𝑧(2nd ‘((1st𝐺)‘𝑌))𝑤)‘𝑓)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑌))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑤))(𝐿𝑧)) = (((⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)(⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑌, 𝑧⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑌, 𝑤⟩))((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼𝑧)⟩)))
140110, 128, 1393eqtr4d 2654 . . 3 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐿𝑤)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑋))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑤))((𝑧(2nd ‘((1st𝐺)‘𝑋))𝑤)‘𝑓)) = (((𝑧(2nd ‘((1st𝐺)‘𝑌))𝑤)‘𝑓)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑌))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑤))(𝐿𝑧)))
141140ralrimivvva 2955 . 2 (𝜑 → ∀𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)((𝐿𝑤)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑋))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑤))((𝑧(2nd ‘((1st𝐺)‘𝑋))𝑤)‘𝑓)) = (((𝑧(2nd ‘((1st𝐺)‘𝑌))𝑤)‘𝑓)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑌))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑤))(𝐿𝑧)))
142 curf2.n . . 3 𝑁 = (𝐷 Nat 𝐸)
1431, 2, 3, 4, 5, 6, 9, 40curf1cl 16691 . . 3 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
1441, 2, 3, 4, 5, 6, 10, 44curf1cl 16691 . . 3 (𝜑 → ((1st𝐺)‘𝑌) ∈ (𝐷 Func 𝐸))
145142, 6, 27, 17, 84, 143, 144isnat2 16431 . 2 (𝜑 → (𝐿 ∈ (((1st𝐺)‘𝑋)𝑁((1st𝐺)‘𝑌)) ↔ (𝐿X𝑧𝐵 (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)) ∧ ∀𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)((𝐿𝑤)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑋))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑤))((𝑧(2nd ‘((1st𝐺)‘𝑋))𝑤)‘𝑓)) = (((𝑧(2nd ‘((1st𝐺)‘𝑌))𝑤)‘𝑓)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑌))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑤))(𝐿𝑧)))))
14656, 141, 145mpbir2and 959 1 (𝜑𝐿 ∈ (((1st𝐺)‘𝑋)𝑁((1st𝐺)‘𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  Rel wrel 5043  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Xcixp 7794  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149   Func cfunc 16337   Nat cnat 16424   ×c cxpc 16631   curryF ccurf 16673 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-func 16341  df-nat 16426  df-xpc 16635  df-curf 16677 This theorem is referenced by:  curfcl  16695
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