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Theorem curf2cl 17561
 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 17559 . . 3 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
14 eqid 2758 . . . . . . . . . 10 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
1514, 2, 6xpcbas 17508 . . . . . . . . 9 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
16 eqid 2758 . . . . . . . . 9 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
17 eqid 2758 . . . . . . . . 9 (Hom ‘𝐸) = (Hom ‘𝐸)
18 relfunc 17205 . . . . . . . . . . 11 Rel ((𝐶 ×c 𝐷) Func 𝐸)
19 1st2ndbr 7751 . . . . . . . . . . 11 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
2018, 5, 19sylancr 590 . . . . . . . . . 10 (𝜑 → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
2120adantr 484 . . . . . . . . 9 ((𝜑𝑧𝐵) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
22 opelxpi 5565 . . . . . . . . . 10 ((𝑋𝐴𝑧𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
239, 22sylan 583 . . . . . . . . 9 ((𝜑𝑧𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
24 opelxpi 5565 . . . . . . . . . 10 ((𝑌𝐴𝑧𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
2510, 24sylan 583 . . . . . . . . 9 ((𝜑𝑧𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
2615, 16, 17, 21, 23, 25funcf2 17211 . . . . . . . 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 2758 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
289adantr 484 . . . . . . . . . 10 ((𝜑𝑧𝐵) → 𝑋𝐴)
29 simpr 488 . . . . . . . . . 10 ((𝜑𝑧𝐵) → 𝑧𝐵)
3010adantr 484 . . . . . . . . . 10 ((𝜑𝑧𝐵) → 𝑌𝐴)
3114, 2, 6, 7, 27, 28, 29, 30, 29, 16xpchom2 17516 . . . . . . . . 9 ((𝜑𝑧𝐵) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
3231feq2d 6489 . . . . . . . 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 235 . . . . . . 7 ((𝜑𝑧𝐵) → (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩):((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧))⟶(((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)))
3411adantr 484 . . . . . . 7 ((𝜑𝑧𝐵) → 𝐾 ∈ (𝑋𝐻𝑌))
354adantr 484 . . . . . . . 8 ((𝜑𝑧𝐵) → 𝐷 ∈ Cat)
366, 27, 8, 35, 29catidcl 17025 . . . . . . 7 ((𝜑𝑧𝐵) → (𝐼𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
3733, 34, 36fovrnd 7322 . . . . . 6 ((𝜑𝑧𝐵) → (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) ∈ (((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)))
383adantr 484 . . . . . . . . 9 ((𝜑𝑧𝐵) → 𝐶 ∈ Cat)
395adantr 484 . . . . . . . . 9 ((𝜑𝑧𝐵) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
40 eqid 2758 . . . . . . . . 9 ((1st𝐺)‘𝑋) = ((1st𝐺)‘𝑋)
411, 2, 38, 35, 39, 6, 28, 40, 29curf11 17556 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = (𝑋(1st𝐹)𝑧))
42 df-ov 7159 . . . . . . . 8 (𝑋(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩)
4341, 42eqtrdi 2809 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
44 eqid 2758 . . . . . . . . 9 ((1st𝐺)‘𝑌) = ((1st𝐺)‘𝑌)
451, 2, 38, 35, 39, 6, 30, 44, 29curf11 17556 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = (𝑌(1st𝐹)𝑧))
46 df-ov 7159 . . . . . . . 8 (𝑌(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩)
4745, 46eqtrdi 2809 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩))
4843, 47oveq12d 7174 . . . . . 6 ((𝜑𝑧𝐵) → (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)) = (((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)))
4937, 48eleqtrrd 2855 . . . . 5 ((𝜑𝑧𝐵) → (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) ∈ (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
5049ralrimiva 3113 . . . 4 (𝜑 → ∀𝑧𝐵 (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) ∈ (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
516fvexi 6677 . . . . 5 𝐵 ∈ V
52 mptelixpg 8530 . . . . 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))))
5351, 52ax-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)))
5450, 53sylibr 237 . . 3 (𝜑 → (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))) ∈ X𝑧𝐵 (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
5513, 54eqeltrd 2852 . 2 (𝜑𝐿X𝑧𝐵 (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
56 eqid 2758 . . . . . . . . . 10 (Id‘𝐶) = (Id‘𝐶)
573adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat)
589adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋𝐴)
59 eqid 2758 . . . . . . . . . 10 (comp‘𝐶) = (comp‘𝐶)
6010adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑌𝐴)
6111adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐾 ∈ (𝑋𝐻𝑌))
622, 7, 56, 57, 58, 59, 60, 61catrid 17027 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐾(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐾)
632, 7, 56, 57, 58, 59, 60, 61catlid 17026 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾) = 𝐾)
6462, 63eqtr4d 2796 . . . . . . . 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))
654adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat)
66 simpr1 1191 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧𝐵)
67 eqid 2758 . . . . . . . . . 10 (comp‘𝐷) = (comp‘𝐷)
68 simpr2 1192 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤𝐵)
69 simpr3 1193 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))
706, 27, 8, 65, 66, 67, 68, 69catlid 17026 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓) = 𝑓)
716, 27, 8, 65, 66, 67, 68, 69catrid 17027 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼𝑧)) = 𝑓)
7270, 71eqtr4d 2796 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓) = (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼𝑧)))
7364, 72opeq12d 4774 . . . . . . 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))⟩)
74 eqid 2758 . . . . . . . 8 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
752, 7, 56, 57, 58catidcl 17025 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
766, 27, 8, 65, 68catidcl 17025 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤))
7714, 2, 6, 7, 27, 58, 66, 58, 68, 59, 67, 74, 60, 68, 75, 69, 61, 76xpcco2 17517 . . . . . . 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)⟩)
78363ad2antr1 1185 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
792, 7, 56, 57, 60catidcl 17025 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐻𝑌))
8014, 2, 6, 7, 27, 58, 66, 60, 66, 59, 67, 74, 60, 68, 61, 78, 79, 69xpcco2 17517 . . . . . . 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))⟩)
8173, 77, 803eqtr4d 2803 . . . . . 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)⟩))
8281fveq2d 6667 . . . . 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)⟩)))
83 eqid 2758 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
8420adantr 484 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
85233ad2antr1 1185 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
8658, 68opelxpd 5566 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
8760, 68opelxpd 5566 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑤⟩ ∈ (𝐴 × 𝐵))
8875, 69opelxpd 5566 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
8914, 2, 6, 7, 27, 58, 66, 58, 68, 16xpchom2 17516 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩) = ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
9088, 89eleqtrrd 2855 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩))
9161, 76opelxpd 5566 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑤)⟩ ∈ ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
9214, 2, 6, 7, 27, 58, 68, 60, 68, 16xpchom2 17516 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
9391, 92eleqtrrd 2855 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑤)⟩ ∈ (⟨𝑋, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩))
9415, 16, 74, 83, 84, 85, 86, 87, 90, 93funcco 17214 . . . . 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⟩)))
95253ad2antr1 1185 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
9661, 78opelxpd 5566 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑧)⟩ ∈ ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
9714, 2, 6, 7, 27, 58, 66, 60, 66, 16xpchom2 17516 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
9896, 97eleqtrrd 2855 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑧)⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩))
9979, 69opelxpd 5566 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
10014, 2, 6, 7, 27, 60, 66, 60, 68, 16xpchom2 17516 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑌, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
10199, 100eleqtrrd 2855 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ (⟨𝑌, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩))
10215, 16, 74, 83, 84, 85, 95, 87, 98, 101funcco 17214 . . . . 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)⟩)))
10382, 94, 1023eqtr3d 2801 . . . 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)⟩)))
1045adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
1051, 2, 57, 65, 104, 6, 58, 40, 66curf11 17556 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = (𝑋(1st𝐹)𝑧))
106105, 42eqtrdi 2809 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
1071, 2, 57, 65, 104, 6, 58, 40, 68curf11 17556 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑤) = (𝑋(1st𝐹)𝑤))
108 df-ov 7159 . . . . . . . 8 (𝑋(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩)
109107, 108eqtrdi 2809 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩))
110106, 109opeq12d 4774 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑋))‘𝑤)⟩ = ⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑋, 𝑤⟩)⟩)
1111, 2, 57, 65, 104, 6, 60, 44, 68curf11 17556 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑤) = (𝑌(1st𝐹)𝑤))
112 df-ov 7159 . . . . . . 7 (𝑌(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑌, 𝑤⟩)
113111, 112eqtrdi 2809 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑤) = ((1st𝐹)‘⟨𝑌, 𝑤⟩))
114110, 113oveq12d 7174 . . . . 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⟩)))
1151, 2, 57, 65, 104, 6, 7, 8, 58, 60, 61, 12, 68curf2val 17560 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑤) = (𝐾(⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)(𝐼𝑤)))
116 df-ov 7159 . . . . . 6 (𝐾(⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)(𝐼𝑤)) = ((⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼𝑤)⟩)
117115, 116eqtrdi 2809 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑤) = ((⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼𝑤)⟩))
1181, 2, 57, 65, 104, 6, 58, 40, 66, 27, 56, 68, 69curf12 17557 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑋))𝑤)‘𝑓) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)𝑓))
119 df-ov 7159 . . . . . 6 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)𝑓) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)
120118, 119eqtrdi 2809 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑋))𝑤)‘𝑓) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩))
121114, 117, 120oveq123d 7177 . . . 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⟩)))
1221, 2, 57, 65, 104, 6, 60, 44, 66curf11 17556 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = (𝑌(1st𝐹)𝑧))
123122, 46eqtrdi 2809 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩))
124106, 123opeq12d 4774 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑌))‘𝑧)⟩ = ⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑌, 𝑧⟩)⟩)
125124, 113oveq12d 7174 . . . . 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⟩)))
1261, 2, 57, 65, 104, 6, 60, 44, 66, 27, 56, 68, 69curf12 17557 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑌))𝑤)‘𝑓) = (((Id‘𝐶)‘𝑌)(⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)𝑓))
127 df-ov 7159 . . . . . 6 (((Id‘𝐶)‘𝑌)(⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)𝑓) = ((⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)
128126, 127eqtrdi 2809 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑌))𝑤)‘𝑓) = ((⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩))
1291, 2, 57, 65, 104, 6, 7, 8, 58, 60, 61, 12, 66curf2val 17560 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑧) = (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))
130 df-ov 7159 . . . . . 6 (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼𝑧)⟩)
131129, 130eqtrdi 2809 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑧) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼𝑧)⟩))
132125, 128, 131oveq123d 7177 . . . 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)⟩)))
133103, 121, 1323eqtr4d 2803 . . 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)))
134133ralrimivvva 3121 . 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)))
135 curf2.n . . 3 𝑁 = (𝐷 Nat 𝐸)
1361, 2, 3, 4, 5, 6, 9, 40curf1cl 17558 . . 3 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
1371, 2, 3, 4, 5, 6, 10, 44curf1cl 17558 . . 3 (𝜑 → ((1st𝐺)‘𝑌) ∈ (𝐷 Func 𝐸))
138135, 6, 27, 17, 83, 136, 137isnat2 17291 . 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)))))
13955, 134, 138mpbir2and 712 1 (𝜑𝐿 ∈ (((1st𝐺)‘𝑋)𝑁((1st𝐺)‘𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409  ⟨cop 4531   class class class wbr 5036   ↦ cmpt 5116   × cxp 5526  Rel wrel 5533  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156  1st c1st 7697  2nd c2nd 7698  Xcixp 8492  Basecbs 16555  Hom chom 16648  compcco 16649  Catccat 17007  Idccid 17008   Func cfunc 17197   Nat cnat 17284   ×c cxpc 17498   curryF ccurf 17540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-er 8305  df-map 8424  df-ixp 8493  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-3 11751  df-4 11752  df-5 11753  df-6 11754  df-7 11755  df-8 11756  df-9 11757  df-n0 11948  df-z 12034  df-dec 12151  df-uz 12296  df-fz 12953  df-struct 16557  df-ndx 16558  df-slot 16559  df-base 16561  df-hom 16661  df-cco 16662  df-cat 17011  df-cid 17012  df-func 17201  df-nat 17286  df-xpc 17502  df-curf 17544 This theorem is referenced by:  curfcl  17562
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