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Theorem curf2cl 16918
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 16916 . . 3 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
14 eqid 2651 . . . . . . . . . 10 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
1514, 2, 6xpcbas 16865 . . . . . . . . 9 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
16 eqid 2651 . . . . . . . . 9 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
17 eqid 2651 . . . . . . . . 9 (Hom ‘𝐸) = (Hom ‘𝐸)
18 relfunc 16569 . . . . . . . . . . 11 Rel ((𝐶 ×c 𝐷) Func 𝐸)
19 1st2ndbr 7261 . . . . . . . . . . 11 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
2018, 5, 19sylancr 696 . . . . . . . . . 10 (𝜑 → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
2120adantr 480 . . . . . . . . 9 ((𝜑𝑧𝐵) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
22 opelxpi 5182 . . . . . . . . . 10 ((𝑋𝐴𝑧𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
239, 22sylan 487 . . . . . . . . 9 ((𝜑𝑧𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
24 opelxpi 5182 . . . . . . . . . 10 ((𝑌𝐴𝑧𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
2510, 24sylan 487 . . . . . . . . 9 ((𝜑𝑧𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
2615, 16, 17, 21, 23, 25funcf2 16575 . . . . . . . 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 2651 . . . . . . . . . 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 16873 . . . . . . . . 9 ((𝜑𝑧𝐵) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
3231feq2d 6069 . . . . . . . 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 222 . . . . . . 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 16390 . . . . . . 7 ((𝜑𝑧𝐵) → (𝐼𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
3733, 34, 36fovrnd 6848 . . . . . 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 2651 . . . . . . . . 9 ((1st𝐺)‘𝑋) = ((1st𝐺)‘𝑋)
411, 2, 38, 35, 39, 6, 28, 40, 29curf11 16913 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = (𝑋(1st𝐹)𝑧))
42 df-ov 6693 . . . . . . . 8 (𝑋(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩)
4341, 42syl6eq 2701 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
44 eqid 2651 . . . . . . . . 9 ((1st𝐺)‘𝑌) = ((1st𝐺)‘𝑌)
451, 2, 38, 35, 39, 6, 30, 44, 29curf11 16913 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = (𝑌(1st𝐹)𝑧))
46 df-ov 6693 . . . . . . . 8 (𝑌(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩)
4745, 46syl6eq 2701 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩))
4843, 47oveq12d 6708 . . . . . 6 ((𝜑𝑧𝐵) → (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)) = (((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)))
4937, 48eleqtrrd 2733 . . . . 5 ((𝜑𝑧𝐵) → (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) ∈ (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
5049ralrimiva 2995 . . . 4 (𝜑 → ∀𝑧𝐵 (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) ∈ (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
51 fvex 6239 . . . . . 6 (Base‘𝐷) ∈ V
526, 51eqeltri 2726 . . . . 5 𝐵 ∈ V
53 mptelixpg 7987 . . . . 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 224 . . 3 (𝜑 → (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))) ∈ X𝑧𝐵 (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
5613, 55eqeltrd 2730 . 2 (𝜑𝐿X𝑧𝐵 (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
57 eqid 2651 . . . . . . . . . 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 2651 . . . . . . . . . 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 16392 . . . . . . . . 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 16391 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾) = 𝐾)
6563, 64eqtr4d 2688 . . . . . . . 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 1087 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧𝐵)
68 eqid 2651 . . . . . . . . . 10 (comp‘𝐷) = (comp‘𝐷)
69 simpr2 1088 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤𝐵)
70 simpr3 1089 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))
716, 27, 8, 66, 67, 68, 69, 70catlid 16391 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓) = 𝑓)
726, 27, 8, 66, 67, 68, 69, 70catrid 16392 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼𝑧)) = 𝑓)
7371, 72eqtr4d 2688 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓) = (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼𝑧)))
7465, 73opeq12d 4441 . . . . . . 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 2651 . . . . . . . 8 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
762, 7, 57, 58, 59catidcl 16390 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
776, 27, 8, 66, 69catidcl 16390 . . . . . . . 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 16874 . . . . . . 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 1246 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
802, 7, 57, 58, 61catidcl 16390 . . . . . . . 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 16874 . . . . . . 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 2695 . . . . . 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 6233 . . . . 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 2651 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
8520adantr 480 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
86233ad2antr1 1246 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
87 opelxpi 5182 . . . . . . 7 ((𝑋𝐴𝑤𝐵) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
8859, 69, 87syl2anc 694 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
89 opelxpi 5182 . . . . . . 7 ((𝑌𝐴𝑤𝐵) → ⟨𝑌, 𝑤⟩ ∈ (𝐴 × 𝐵))
9061, 69, 89syl2anc 694 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑤⟩ ∈ (𝐴 × 𝐵))
91 opelxpi 5182 . . . . . . . 8 ((((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
9276, 70, 91syl2anc 694 . . . . . . 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 16873 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩) = ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
9492, 93eleqtrrd 2733 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩))
95 opelxpi 5182 . . . . . . . 8 ((𝐾 ∈ (𝑋𝐻𝑌) ∧ (𝐼𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤)) → ⟨𝐾, (𝐼𝑤)⟩ ∈ ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
9662, 77, 95syl2anc 694 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑤)⟩ ∈ ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
9714, 2, 6, 7, 27, 59, 69, 61, 69, 16xpchom2 16873 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
9896, 97eleqtrrd 2733 . . . . . 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 16578 . . . . 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 1246 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
101 opelxpi 5182 . . . . . . . 8 ((𝐾 ∈ (𝑋𝐻𝑌) ∧ (𝐼𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧)) → ⟨𝐾, (𝐼𝑧)⟩ ∈ ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
10262, 79, 101syl2anc 694 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑧)⟩ ∈ ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
10314, 2, 6, 7, 27, 59, 67, 61, 67, 16xpchom2 16873 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
104102, 103eleqtrrd 2733 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑧)⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩))
105 opelxpi 5182 . . . . . . . 8 ((((Id‘𝐶)‘𝑌) ∈ (𝑌𝐻𝑌) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
10680, 70, 105syl2anc 694 . . . . . . 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 16873 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑌, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
108106, 107eleqtrrd 2733 . . . . . 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 16578 . . . . 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 2693 . . . 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 16913 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = (𝑋(1st𝐹)𝑧))
113112, 42syl6eq 2701 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
1141, 2, 58, 66, 111, 6, 59, 40, 69curf11 16913 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑤) = (𝑋(1st𝐹)𝑤))
115 df-ov 6693 . . . . . . . 8 (𝑋(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩)
116114, 115syl6eq 2701 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩))
117113, 116opeq12d 4441 . . . . . 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 16913 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑤) = (𝑌(1st𝐹)𝑤))
119 df-ov 6693 . . . . . . 7 (𝑌(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑌, 𝑤⟩)
120118, 119syl6eq 2701 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑤) = ((1st𝐹)‘⟨𝑌, 𝑤⟩))
121117, 120oveq12d 6708 . . . . 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 16917 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑤) = (𝐾(⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)(𝐼𝑤)))
123 df-ov 6693 . . . . . 6 (𝐾(⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)(𝐼𝑤)) = ((⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼𝑤)⟩)
124122, 123syl6eq 2701 . . . . 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 16914 . . . . . 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 6693 . . . . . 6 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)𝑓) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)
127125, 126syl6eq 2701 . . . . 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 6711 . . . 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 16913 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = (𝑌(1st𝐹)𝑧))
130129, 46syl6eq 2701 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩))
131113, 130opeq12d 4441 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑌))‘𝑧)⟩ = ⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑌, 𝑧⟩)⟩)
132131, 120oveq12d 6708 . . . . 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 16914 . . . . . 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 6693 . . . . . 6 (((Id‘𝐶)‘𝑌)(⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)𝑓) = ((⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)
135133, 134syl6eq 2701 . . . . 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 16917 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑧) = (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))
137 df-ov 6693 . . . . . 6 (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼𝑧)⟩)
138136, 137syl6eq 2701 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑧) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼𝑧)⟩))
139132, 135, 138oveq123d 6711 . . . 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 2695 . . 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 3001 . 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 16915 . . 3 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
1441, 2, 3, 4, 5, 6, 10, 44curf1cl 16915 . . 3 (𝜑 → ((1st𝐺)‘𝑌) ∈ (𝐷 Func 𝐸))
145142, 6, 27, 17, 84, 143, 144isnat2 16655 . 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 977 1 (𝜑𝐿 ∈ (((1st𝐺)‘𝑋)𝑁((1st𝐺)‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cop 4216   class class class wbr 4685  cmpt 4762   × cxp 5141  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Xcixp 7950  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373   Func cfunc 16561   Nat cnat 16648   ×c cxpc 16855   curryF ccurf 16897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-func 16565  df-nat 16650  df-xpc 16859  df-curf 16901
This theorem is referenced by:  curfcl  16919
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