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Theorem curf2cl 17140
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 17138 . . 3 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
14 eqid 2765 . . . . . . . . . 10 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
1514, 2, 6xpcbas 17087 . . . . . . . . 9 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
16 eqid 2765 . . . . . . . . 9 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
17 eqid 2765 . . . . . . . . 9 (Hom ‘𝐸) = (Hom ‘𝐸)
18 relfunc 16790 . . . . . . . . . . 11 Rel ((𝐶 ×c 𝐷) Func 𝐸)
19 1st2ndbr 7419 . . . . . . . . . . 11 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
2018, 5, 19sylancr 581 . . . . . . . . . 10 (𝜑 → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
2120adantr 472 . . . . . . . . 9 ((𝜑𝑧𝐵) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
22 opelxpi 5316 . . . . . . . . . 10 ((𝑋𝐴𝑧𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
239, 22sylan 575 . . . . . . . . 9 ((𝜑𝑧𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
24 opelxpi 5316 . . . . . . . . . 10 ((𝑌𝐴𝑧𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
2510, 24sylan 575 . . . . . . . . 9 ((𝜑𝑧𝐵) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
2615, 16, 17, 21, 23, 25funcf2 16796 . . . . . . . 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 2765 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
289adantr 472 . . . . . . . . . 10 ((𝜑𝑧𝐵) → 𝑋𝐴)
29 simpr 477 . . . . . . . . . 10 ((𝜑𝑧𝐵) → 𝑧𝐵)
3010adantr 472 . . . . . . . . . 10 ((𝜑𝑧𝐵) → 𝑌𝐴)
3114, 2, 6, 7, 27, 28, 29, 30, 29, 16xpchom2 17095 . . . . . . . . 9 ((𝜑𝑧𝐵) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
3231feq2d 6211 . . . . . . . 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 223 . . . . . . 7 ((𝜑𝑧𝐵) → (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩):((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧))⟶(((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)))
3411adantr 472 . . . . . . 7 ((𝜑𝑧𝐵) → 𝐾 ∈ (𝑋𝐻𝑌))
354adantr 472 . . . . . . . 8 ((𝜑𝑧𝐵) → 𝐷 ∈ Cat)
366, 27, 8, 35, 29catidcl 16611 . . . . . . 7 ((𝜑𝑧𝐵) → (𝐼𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
3733, 34, 36fovrnd 7006 . . . . . 6 ((𝜑𝑧𝐵) → (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) ∈ (((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)))
383adantr 472 . . . . . . . . 9 ((𝜑𝑧𝐵) → 𝐶 ∈ Cat)
395adantr 472 . . . . . . . . 9 ((𝜑𝑧𝐵) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
40 eqid 2765 . . . . . . . . 9 ((1st𝐺)‘𝑋) = ((1st𝐺)‘𝑋)
411, 2, 38, 35, 39, 6, 28, 40, 29curf11 17135 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = (𝑋(1st𝐹)𝑧))
42 df-ov 6847 . . . . . . . 8 (𝑋(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩)
4341, 42syl6eq 2815 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
44 eqid 2765 . . . . . . . . 9 ((1st𝐺)‘𝑌) = ((1st𝐺)‘𝑌)
451, 2, 38, 35, 39, 6, 30, 44, 29curf11 17135 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = (𝑌(1st𝐹)𝑧))
46 df-ov 6847 . . . . . . . 8 (𝑌(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩)
4745, 46syl6eq 2815 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩))
4843, 47oveq12d 6862 . . . . . 6 ((𝜑𝑧𝐵) → (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)) = (((1st𝐹)‘⟨𝑋, 𝑧⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑌, 𝑧⟩)))
4937, 48eleqtrrd 2847 . . . . 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 6391 . . . . 5 𝐵 ∈ V
52 mptelixpg 8152 . . . . 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 225 . . 3 (𝜑 → (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))) ∈ X𝑧𝐵 (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
5513, 54eqeltrd 2844 . 2 (𝜑𝐿X𝑧𝐵 (((1st ‘((1st𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑌))‘𝑧)))
56 eqid 2765 . . . . . . . . . 10 (Id‘𝐶) = (Id‘𝐶)
573adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat)
589adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋𝐴)
59 eqid 2765 . . . . . . . . . 10 (comp‘𝐶) = (comp‘𝐶)
6010adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑌𝐴)
6111adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐾 ∈ (𝑋𝐻𝑌))
622, 7, 56, 57, 58, 59, 60, 61catrid 16613 . . . . . . . . 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 16612 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐾) = 𝐾)
6462, 63eqtr4d 2802 . . . . . . . 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 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat)
66 simpr1 1248 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧𝐵)
67 eqid 2765 . . . . . . . . . 10 (comp‘𝐷) = (comp‘𝐷)
68 simpr2 1250 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤𝐵)
69 simpr3 1252 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))
706, 27, 8, 65, 66, 67, 68, 69catlid 16612 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓) = 𝑓)
716, 27, 8, 65, 66, 67, 68, 69catrid 16613 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼𝑧)) = 𝑓)
7270, 71eqtr4d 2802 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼𝑤)(⟨𝑧, 𝑤⟩(comp‘𝐷)𝑤)𝑓) = (𝑓(⟨𝑧, 𝑧⟩(comp‘𝐷)𝑤)(𝐼𝑧)))
7364, 72opeq12d 4569 . . . . . . 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 2765 . . . . . . . 8 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
752, 7, 56, 57, 58catidcl 16611 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
766, 27, 8, 65, 68catidcl 16611 . . . . . . . 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 17096 . . . . . . 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 1239 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
792, 7, 56, 57, 60catidcl 16611 . . . . . . . 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 17096 . . . . . . 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 2809 . . . . . 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 6381 . . . . 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 2765 . . . . . 6 (comp‘𝐸) = (comp‘𝐸)
8420adantr 472 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
85233ad2antr1 1239 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
86 opelxpi 5316 . . . . . . 7 ((𝑋𝐴𝑤𝐵) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
8758, 68, 86syl2anc 579 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
88 opelxpi 5316 . . . . . . 7 ((𝑌𝐴𝑤𝐵) → ⟨𝑌, 𝑤⟩ ∈ (𝐴 × 𝐵))
8960, 68, 88syl2anc 579 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑤⟩ ∈ (𝐴 × 𝐵))
90 opelxpi 5316 . . . . . . . 8 ((((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
9175, 69, 90syl2anc 579 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
9214, 2, 6, 7, 27, 58, 66, 58, 68, 16xpchom2 17095 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩) = ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
9391, 92eleqtrrd 2847 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑓⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩))
94 opelxpi 5316 . . . . . . . 8 ((𝐾 ∈ (𝑋𝐻𝑌) ∧ (𝐼𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤)) → ⟨𝐾, (𝐼𝑤)⟩ ∈ ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
9561, 76, 94syl2anc 579 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑤)⟩ ∈ ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
9614, 2, 6, 7, 27, 58, 68, 60, 68, 16xpchom2 17095 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤)))
9795, 96eleqtrrd 2847 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑤)⟩ ∈ (⟨𝑋, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩))
9815, 16, 74, 83, 84, 85, 87, 89, 93, 97funcco 16799 . . . . 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⟩)))
99253ad2antr1 1239 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑌, 𝑧⟩ ∈ (𝐴 × 𝐵))
100 opelxpi 5316 . . . . . . . 8 ((𝐾 ∈ (𝑋𝐻𝑌) ∧ (𝐼𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧)) → ⟨𝐾, (𝐼𝑧)⟩ ∈ ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
10161, 78, 100syl2anc 579 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑧)⟩ ∈ ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
10214, 2, 6, 7, 27, 58, 66, 60, 66, 16xpchom2 17095 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧)))
103101, 102eleqtrrd 2847 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝐾, (𝐼𝑧)⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑧⟩))
104 opelxpi 5316 . . . . . . . 8 ((((Id‘𝐶)‘𝑌) ∈ (𝑌𝐻𝑌) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
10579, 69, 104syl2anc 579 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
10614, 2, 6, 7, 27, 60, 66, 60, 68, 16xpchom2 17095 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑌, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩) = ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤)))
107105, 106eleqtrrd 2847 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑌), 𝑓⟩ ∈ (⟨𝑌, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑤⟩))
10815, 16, 74, 83, 84, 85, 99, 89, 103, 107funcco 16799 . . . . 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)⟩)))
10982, 98, 1083eqtr3d 2807 . . . 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)⟩)))
1105adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
1111, 2, 57, 65, 110, 6, 58, 40, 66curf11 17135 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = (𝑋(1st𝐹)𝑧))
112111, 42syl6eq 2815 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
1131, 2, 57, 65, 110, 6, 58, 40, 68curf11 17135 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑤) = (𝑋(1st𝐹)𝑤))
114 df-ov 6847 . . . . . . . 8 (𝑋(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩)
115113, 114syl6eq 2815 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑋))‘𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩))
116112, 115opeq12d 4569 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑋))‘𝑤)⟩ = ⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑋, 𝑤⟩)⟩)
1171, 2, 57, 65, 110, 6, 60, 44, 68curf11 17135 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑤) = (𝑌(1st𝐹)𝑤))
118 df-ov 6847 . . . . . . 7 (𝑌(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑌, 𝑤⟩)
119117, 118syl6eq 2815 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑤) = ((1st𝐹)‘⟨𝑌, 𝑤⟩))
120116, 119oveq12d 6862 . . . . 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⟩)))
1211, 2, 57, 65, 110, 6, 7, 8, 58, 60, 61, 12, 68curf2val 17139 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑤) = (𝐾(⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)(𝐼𝑤)))
122 df-ov 6847 . . . . . 6 (𝐾(⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)(𝐼𝑤)) = ((⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼𝑤)⟩)
123121, 122syl6eq 2815 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑤) = ((⟨𝑋, 𝑤⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨𝐾, (𝐼𝑤)⟩))
1241, 2, 57, 65, 110, 6, 58, 40, 66, 27, 56, 68, 69curf12 17136 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑋))𝑤)‘𝑓) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)𝑓))
125 df-ov 6847 . . . . . 6 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)𝑓) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩)
126124, 125syl6eq 2815 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑋))𝑤)‘𝑓) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑓⟩))
127120, 123, 126oveq123d 6865 . . . 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⟩)))
1281, 2, 57, 65, 110, 6, 60, 44, 66curf11 17135 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = (𝑌(1st𝐹)𝑧))
129128, 46syl6eq 2815 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘((1st𝐺)‘𝑌))‘𝑧) = ((1st𝐹)‘⟨𝑌, 𝑧⟩))
130112, 129opeq12d 4569 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1st ‘((1st𝐺)‘𝑋))‘𝑧), ((1st ‘((1st𝐺)‘𝑌))‘𝑧)⟩ = ⟨((1st𝐹)‘⟨𝑋, 𝑧⟩), ((1st𝐹)‘⟨𝑌, 𝑧⟩)⟩)
131130, 119oveq12d 6862 . . . . 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⟩)))
1321, 2, 57, 65, 110, 6, 60, 44, 66, 27, 56, 68, 69curf12 17136 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑌))𝑤)‘𝑓) = (((Id‘𝐶)‘𝑌)(⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)𝑓))
133 df-ov 6847 . . . . . 6 (((Id‘𝐶)‘𝑌)(⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)𝑓) = ((⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩)
134132, 133syl6eq 2815 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st𝐺)‘𝑌))𝑤)‘𝑓) = ((⟨𝑌, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑌), 𝑓⟩))
1351, 2, 57, 65, 110, 6, 7, 8, 58, 60, 61, 12, 66curf2val 17139 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑧) = (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))
136 df-ov 6847 . . . . . 6 (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼𝑧)⟩)
137135, 136syl6eq 2815 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿𝑧) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)‘⟨𝐾, (𝐼𝑧)⟩))
138131, 134, 137oveq123d 6865 . . . 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)⟩)))
139109, 127, 1383eqtr4d 2809 . . 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)))
140139ralrimivvva 3119 . 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)))
141 curf2.n . . 3 𝑁 = (𝐷 Nat 𝐸)
1421, 2, 3, 4, 5, 6, 9, 40curf1cl 17137 . . 3 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
1431, 2, 3, 4, 5, 6, 10, 44curf1cl 17137 . . 3 (𝜑 → ((1st𝐺)‘𝑌) ∈ (𝐷 Func 𝐸))
144141, 6, 27, 17, 83, 142, 143isnat2 16876 . 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)))))
14555, 140, 144mpbir2and 704 1 (𝜑𝐿 ∈ (((1st𝐺)‘𝑋)𝑁((1st𝐺)‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cop 4342   class class class wbr 4811  cmpt 4890   × cxp 5277  Rel wrel 5284  wf 6066  cfv 6070  (class class class)co 6844  1st c1st 7366  2nd c2nd 7367  Xcixp 8115  Basecbs 16133  Hom chom 16228  compcco 16229  Catccat 16593  Idccid 16594   Func cfunc 16782   Nat cnat 16869   ×c cxpc 17077   curryF ccurf 17119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-oadd 7770  df-er 7949  df-map 8064  df-ixp 8116  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-nn 11277  df-2 11337  df-3 11338  df-4 11339  df-5 11340  df-6 11341  df-7 11342  df-8 11343  df-9 11344  df-n0 11541  df-z 11627  df-dec 11744  df-uz 11890  df-fz 12537  df-struct 16135  df-ndx 16136  df-slot 16137  df-base 16139  df-hom 16241  df-cco 16242  df-cat 16597  df-cid 16598  df-func 16786  df-nat 16871  df-xpc 17081  df-curf 17123
This theorem is referenced by:  curfcl  17141
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