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Theorem curfcl 17065
Description: The curry functor of a functor 𝐹:𝐶 × 𝐷𝐸 is a functor curryF (𝐹):𝐶⟶(𝐷𝐸). (Contributed by Mario Carneiro, 13-Jan-2017.)
Hypotheses
Ref Expression
curfcl.g 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curfcl.q 𝑄 = (𝐷 FuncCat 𝐸)
curfcl.c (𝜑𝐶 ∈ Cat)
curfcl.d (𝜑𝐷 ∈ Cat)
curfcl.f (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
Assertion
Ref Expression
curfcl (𝜑𝐺 ∈ (𝐶 Func 𝑄))

Proof of Theorem curfcl
Dummy variables 𝑤 𝑔 𝑥 𝑦 𝑧 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curfcl.g . . . 4 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2 eqid 2752 . . . 4 (Base‘𝐶) = (Base‘𝐶)
3 curfcl.c . . . 4 (𝜑𝐶 ∈ Cat)
4 curfcl.d . . . 4 (𝜑𝐷 ∈ Cat)
5 curfcl.f . . . 4 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6 eqid 2752 . . . 4 (Base‘𝐷) = (Base‘𝐷)
7 eqid 2752 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
8 eqid 2752 . . . 4 (Id‘𝐶) = (Id‘𝐶)
9 eqid 2752 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
10 eqid 2752 . . . 4 (Id‘𝐷) = (Id‘𝐷)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curfval 17056 . . 3 (𝜑𝐺 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩)
12 fvex 6354 . . . . . . 7 (Base‘𝐶) ∈ V
1312mptex 6642 . . . . . 6 (𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) ∈ V
1412, 12mpt2ex 7407 . . . . . 6 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) ∈ V
1513, 14op1std 7335 . . . . 5 (𝐺 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩ → (1st𝐺) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
1611, 15syl 17 . . . 4 (𝜑 → (1st𝐺) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
1713, 14op2ndd 7336 . . . . 5 (𝐺 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩ → (2nd𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))))
1811, 17syl 17 . . . 4 (𝜑 → (2nd𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))))
1916, 18opeq12d 4553 . . 3 (𝜑 → ⟨(1st𝐺), (2nd𝐺)⟩ = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩)
2011, 19eqtr4d 2789 . 2 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
21 curfcl.q . . . . 5 𝑄 = (𝐷 FuncCat 𝐸)
2221fucbas 16813 . . . 4 (𝐷 Func 𝐸) = (Base‘𝑄)
23 eqid 2752 . . . . 5 (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
2421, 23fuchom 16814 . . . 4 (𝐷 Nat 𝐸) = (Hom ‘𝑄)
25 eqid 2752 . . . 4 (Id‘𝑄) = (Id‘𝑄)
26 eqid 2752 . . . 4 (comp‘𝐶) = (comp‘𝐶)
27 eqid 2752 . . . 4 (comp‘𝑄) = (comp‘𝑄)
28 funcrcl 16716 . . . . . . 7 (𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸) → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
295, 28syl 17 . . . . . 6 (𝜑 → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
3029simprd 482 . . . . 5 (𝜑𝐸 ∈ Cat)
3121, 4, 30fuccat 16823 . . . 4 (𝜑𝑄 ∈ Cat)
32 opex 5073 . . . . . 6 ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ ∈ V
3332a1i 11 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ ∈ V)
343adantr 472 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
354adantr 472 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
365adantr 472 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
37 simpr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
38 eqid 2752 . . . . . 6 ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑥)
391, 2, 34, 35, 36, 6, 37, 38curf1cl 17061 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
4033, 16, 39fmpt2d 6548 . . . 4 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(𝐷 Func 𝐸))
41 eqid 2752 . . . . . 6 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))
42 ovex 6833 . . . . . . 7 (𝑥(Hom ‘𝐶)𝑦) ∈ V
4342mptex 6642 . . . . . 6 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))) ∈ V
4441, 43fnmpt2i 7399 . . . . 5 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) Fn ((Base‘𝐶) × (Base‘𝐶))
4518fneq1d 6134 . . . . 5 (𝜑 → ((2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) Fn ((Base‘𝐶) × (Base‘𝐶))))
4644, 45mpbiri 248 . . . 4 (𝜑 → (2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)))
47 fvex 6354 . . . . . . 7 (Base‘𝐷) ∈ V
4847mptex 6642 . . . . . 6 (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))) ∈ V
4948a1i 11 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))) ∈ V)
5018oveqd 6822 . . . . . 6 (𝜑 → (𝑥(2nd𝐺)𝑦) = (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))𝑦))
5141ovmpt4g 6940 . . . . . . 7 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))) ∈ V) → (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))𝑦) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))
5243, 51mp3an3 1554 . . . . . 6 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))𝑦) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))
5350, 52sylan9eq 2806 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))
543ad2antrr 764 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐶 ∈ Cat)
554ad2antrr 764 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
565ad2antrr 764 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
57 simplrl 819 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
58 simplrr 820 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
59 simpr 479 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
60 eqid 2752 . . . . . 6 ((𝑥(2nd𝐺)𝑦)‘𝑔) = ((𝑥(2nd𝐺)𝑦)‘𝑔)
611, 2, 54, 55, 56, 6, 9, 10, 57, 58, 59, 60, 23curf2cl 17064 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐺)𝑦)‘𝑔) ∈ (((1st𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st𝐺)‘𝑦)))
6249, 53, 61fmpt2d 6548 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st𝐺)‘𝑦)))
63 eqid 2752 . . . . . . . . . 10 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
6463, 2, 6xpcbas 17011 . . . . . . . . 9 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×c 𝐷))
65 eqid 2752 . . . . . . . . 9 (Id‘(𝐶 ×c 𝐷)) = (Id‘(𝐶 ×c 𝐷))
66 eqid 2752 . . . . . . . . 9 (Id‘𝐸) = (Id‘𝐸)
67 relfunc 16715 . . . . . . . . . . 11 Rel ((𝐶 ×c 𝐷) Func 𝐸)
68 1st2ndbr 7376 . . . . . . . . . . 11 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
6967, 5, 68sylancr 698 . . . . . . . . . 10 (𝜑 → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
7069ad2antrr 764 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
71 opelxpi 5297 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
7271adantll 752 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
7364, 65, 66, 70, 72funcid 16723 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑥, 𝑦⟩)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑥, 𝑦⟩)))
743ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat)
754ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
7637adantr 472 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
77 simpr 479 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐷))
7863, 74, 75, 2, 6, 8, 10, 65, 76, 77xpcid 17022 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((Id‘(𝐶 ×c 𝐷))‘⟨𝑥, 𝑦⟩) = ⟨((Id‘𝐶)‘𝑥), ((Id‘𝐷)‘𝑦)⟩)
7978fveq2d 6348 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑥, 𝑦⟩)) = ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑥), ((Id‘𝐷)‘𝑦)⟩))
80 df-ov 6808 . . . . . . . . 9 (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑥), ((Id‘𝐷)‘𝑦)⟩)
8179, 80syl6eqr 2804 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑥, 𝑦⟩)) = (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦)))
825ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
831, 2, 74, 75, 82, 6, 76, 38, 77curf11 17059 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑦) = (𝑥(1st𝐹)𝑦))
84 df-ov 6808 . . . . . . . . . 10 (𝑥(1st𝐹)𝑦) = ((1st𝐹)‘⟨𝑥, 𝑦⟩)
8583, 84syl6req 2803 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st𝐹)‘⟨𝑥, 𝑦⟩) = ((1st ‘((1st𝐺)‘𝑥))‘𝑦))
8685fveq2d 6348 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((Id‘𝐸)‘((1st𝐹)‘⟨𝑥, 𝑦⟩)) = ((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑦)))
8773, 81, 863eqtr3d 2794 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑦)))
8887mpteq2dva 4888 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦))) = (𝑦 ∈ (Base‘𝐷) ↦ ((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑦))))
8930adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
90 eqid 2752 . . . . . . . . . 10 (Base‘𝐸) = (Base‘𝐸)
9190, 66cidfn 16533 . . . . . . . . 9 (𝐸 ∈ Cat → (Id‘𝐸) Fn (Base‘𝐸))
9289, 91syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (Id‘𝐸) Fn (Base‘𝐸))
93 dffn2 6200 . . . . . . . 8 ((Id‘𝐸) Fn (Base‘𝐸) ↔ (Id‘𝐸):(Base‘𝐸)⟶V)
9492, 93sylib 208 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (Id‘𝐸):(Base‘𝐸)⟶V)
95 relfunc 16715 . . . . . . . . 9 Rel (𝐷 Func 𝐸)
96 1st2ndbr 7376 . . . . . . . . 9 ((Rel (𝐷 Func 𝐸) ∧ ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
9795, 39, 96sylancr 698 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
986, 90, 97funcf1 16719 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
99 fcompt 6555 . . . . . . 7 (((Id‘𝐸):(Base‘𝐸)⟶V ∧ (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸)) → ((Id‘𝐸) ∘ (1st ‘((1st𝐺)‘𝑥))) = (𝑦 ∈ (Base‘𝐷) ↦ ((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑦))))
10094, 98, 99syl2anc 696 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐸) ∘ (1st ‘((1st𝐺)‘𝑥))) = (𝑦 ∈ (Base‘𝐷) ↦ ((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑦))))
10188, 100eqtr4d 2789 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦))) = ((Id‘𝐸) ∘ (1st ‘((1st𝐺)‘𝑥))))
1022, 9, 8, 34, 37catidcl 16536 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
103 eqid 2752 . . . . . 6 ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥))
1041, 2, 34, 35, 36, 6, 9, 10, 37, 37, 102, 103curf2 17062 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = (𝑦 ∈ (Base‘𝐷) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦))))
10521, 25, 66, 39fucid 16824 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑄)‘((1st𝐺)‘𝑥)) = ((Id‘𝐸) ∘ (1st ‘((1st𝐺)‘𝑥))))
106101, 104, 1053eqtr4d 2796 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝑄)‘((1st𝐺)‘𝑥)))
10733ad2ant1 1127 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
108107adantr 472 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat)
10943ad2ant1 1127 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
110109adantr 472 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
11153ad2ant1 1127 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
112111adantr 472 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
113 simp21 1246 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ (Base‘𝐶))
114113adantr 472 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
115 simpr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑤 ∈ (Base‘𝐷))
1161, 2, 108, 110, 112, 6, 114, 38, 115curf11 17059 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑤) = (𝑥(1st𝐹)𝑤))
117 df-ov 6808 . . . . . . . . . . 11 (𝑥(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑥, 𝑤⟩)
118116, 117syl6eq 2802 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑤) = ((1st𝐹)‘⟨𝑥, 𝑤⟩))
119 simp22 1247 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ (Base‘𝐶))
120119adantr 472 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐶))
121 eqid 2752 . . . . . . . . . . . 12 ((1st𝐺)‘𝑦) = ((1st𝐺)‘𝑦)
1221, 2, 108, 110, 112, 6, 120, 121, 115curf11 17059 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑦))‘𝑤) = (𝑦(1st𝐹)𝑤))
123 df-ov 6808 . . . . . . . . . . 11 (𝑦(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑦, 𝑤⟩)
124122, 123syl6eq 2802 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑦))‘𝑤) = ((1st𝐹)‘⟨𝑦, 𝑤⟩))
125118, 124opeq12d 4553 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨((1st ‘((1st𝐺)‘𝑥))‘𝑤), ((1st ‘((1st𝐺)‘𝑦))‘𝑤)⟩ = ⟨((1st𝐹)‘⟨𝑥, 𝑤⟩), ((1st𝐹)‘⟨𝑦, 𝑤⟩)⟩)
126 simp23 1248 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
127126adantr 472 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑧 ∈ (Base‘𝐶))
128 eqid 2752 . . . . . . . . . . 11 ((1st𝐺)‘𝑧) = ((1st𝐺)‘𝑧)
1291, 2, 108, 110, 112, 6, 127, 128, 115curf11 17059 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑧))‘𝑤) = (𝑧(1st𝐹)𝑤))
130 df-ov 6808 . . . . . . . . . 10 (𝑧(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑧, 𝑤⟩)
131129, 130syl6eq 2802 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑧))‘𝑤) = ((1st𝐹)‘⟨𝑧, 𝑤⟩))
132125, 131oveq12d 6823 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (⟨((1st ‘((1st𝐺)‘𝑥))‘𝑤), ((1st ‘((1st𝐺)‘𝑦))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑧))‘𝑤)) = (⟨((1st𝐹)‘⟨𝑥, 𝑤⟩), ((1st𝐹)‘⟨𝑦, 𝑤⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑧, 𝑤⟩)))
133 simp3r 1242 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
134133adantr 472 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
135 eqid 2752 . . . . . . . . . 10 ((𝑦(2nd𝐺)𝑧)‘𝑔) = ((𝑦(2nd𝐺)𝑧)‘𝑔)
1361, 2, 108, 110, 112, 6, 9, 10, 120, 127, 134, 135, 115curf2val 17063 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (((𝑦(2nd𝐺)𝑧)‘𝑔)‘𝑤) = (𝑔(⟨𝑦, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)))
137 df-ov 6808 . . . . . . . . 9 (𝑔(⟨𝑦, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑦, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑔, ((Id‘𝐷)‘𝑤)⟩)
138136, 137syl6eq 2802 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (((𝑦(2nd𝐺)𝑧)‘𝑔)‘𝑤) = ((⟨𝑦, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑔, ((Id‘𝐷)‘𝑤)⟩))
139 simp3l 1241 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
140139adantr 472 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
141 eqid 2752 . . . . . . . . . 10 ((𝑥(2nd𝐺)𝑦)‘𝑓) = ((𝑥(2nd𝐺)𝑦)‘𝑓)
1421, 2, 108, 110, 112, 6, 9, 10, 114, 120, 140, 141, 115curf2val 17063 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (((𝑥(2nd𝐺)𝑦)‘𝑓)‘𝑤) = (𝑓(⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑦, 𝑤⟩)((Id‘𝐷)‘𝑤)))
143 df-ov 6808 . . . . . . . . 9 (𝑓(⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑦, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑦, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)
144142, 143syl6eq 2802 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (((𝑥(2nd𝐺)𝑦)‘𝑓)‘𝑤) = ((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑦, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩))
145132, 138, 144oveq123d 6826 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((((𝑦(2nd𝐺)𝑧)‘𝑔)‘𝑤)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑤), ((1st ‘((1st𝐺)‘𝑦))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑧))‘𝑤))(((𝑥(2nd𝐺)𝑦)‘𝑓)‘𝑤)) = (((⟨𝑦, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑔, ((Id‘𝐷)‘𝑤)⟩)(⟨((1st𝐹)‘⟨𝑥, 𝑤⟩), ((1st𝐹)‘⟨𝑦, 𝑤⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑧, 𝑤⟩))((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑦, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)))
146 eqid 2752 . . . . . . . 8 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
147 eqid 2752 . . . . . . . 8 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
148 eqid 2752 . . . . . . . 8 (comp‘𝐸) = (comp‘𝐸)
14967, 112, 68sylancr 698 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
150 opelxpi 5297 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
151113, 150sylan 489 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
152 opelxpi 5297 . . . . . . . . 9 ((𝑦 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑦, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
153119, 152sylan 489 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑦, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
154 opelxpi 5297 . . . . . . . . 9 ((𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
155126, 154sylan 489 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
1566, 7, 10, 110, 115catidcl 16536 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((Id‘𝐷)‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤))
157 opelxpi 5297 . . . . . . . . . 10 ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((Id‘𝐷)‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤)) → ⟨𝑓, ((Id‘𝐷)‘𝑤)⟩ ∈ ((𝑥(Hom ‘𝐶)𝑦) × (𝑤(Hom ‘𝐷)𝑤)))
158140, 156, 157syl2anc 696 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑓, ((Id‘𝐷)‘𝑤)⟩ ∈ ((𝑥(Hom ‘𝐶)𝑦) × (𝑤(Hom ‘𝐷)𝑤)))
15963, 2, 6, 9, 7, 114, 115, 120, 115, 146xpchom2 17019 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (⟨𝑥, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑤⟩) = ((𝑥(Hom ‘𝐶)𝑦) × (𝑤(Hom ‘𝐷)𝑤)))
160158, 159eleqtrrd 2834 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑓, ((Id‘𝐷)‘𝑤)⟩ ∈ (⟨𝑥, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑤⟩))
161 opelxpi 5297 . . . . . . . . . 10 ((𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ ((Id‘𝐷)‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤)) → ⟨𝑔, ((Id‘𝐷)‘𝑤)⟩ ∈ ((𝑦(Hom ‘𝐶)𝑧) × (𝑤(Hom ‘𝐷)𝑤)))
162134, 156, 161syl2anc 696 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑔, ((Id‘𝐷)‘𝑤)⟩ ∈ ((𝑦(Hom ‘𝐶)𝑧) × (𝑤(Hom ‘𝐷)𝑤)))
16363, 2, 6, 9, 7, 120, 115, 127, 115, 146xpchom2 17019 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (⟨𝑦, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩) = ((𝑦(Hom ‘𝐶)𝑧) × (𝑤(Hom ‘𝐷)𝑤)))
164162, 163eleqtrrd 2834 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑔, ((Id‘𝐷)‘𝑤)⟩ ∈ (⟨𝑦, 𝑤⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩))
16564, 146, 147, 148, 149, 151, 153, 155, 160, 164funcco 16724 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑔, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑤⟩, ⟨𝑦, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)) = (((⟨𝑦, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑔, ((Id‘𝐷)‘𝑤)⟩)(⟨((1st𝐹)‘⟨𝑥, 𝑤⟩), ((1st𝐹)‘⟨𝑦, 𝑤⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑧, 𝑤⟩))((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑦, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)))
166 eqid 2752 . . . . . . . . . . 11 (comp‘𝐷) = (comp‘𝐷)
16763, 2, 6, 9, 7, 114, 115, 120, 115, 26, 166, 147, 127, 115, 140, 156, 134, 156xpcco2 17020 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (⟨𝑔, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑤⟩, ⟨𝑦, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨𝑓, ((Id‘𝐷)‘𝑤)⟩) = ⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), (((Id‘𝐷)‘𝑤)(⟨𝑤, 𝑤⟩(comp‘𝐷)𝑤)((Id‘𝐷)‘𝑤))⟩)
1686, 7, 10, 110, 115, 166, 115, 156catlid 16537 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (((Id‘𝐷)‘𝑤)(⟨𝑤, 𝑤⟩(comp‘𝐷)𝑤)((Id‘𝐷)‘𝑤)) = ((Id‘𝐷)‘𝑤))
169168opeq2d 4552 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), (((Id‘𝐷)‘𝑤)(⟨𝑤, 𝑤⟩(comp‘𝐷)𝑤)((Id‘𝐷)‘𝑤))⟩ = ⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), ((Id‘𝐷)‘𝑤)⟩)
170167, 169eqtrd 2786 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → (⟨𝑔, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑤⟩, ⟨𝑦, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨𝑓, ((Id‘𝐷)‘𝑤)⟩) = ⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), ((Id‘𝐷)‘𝑤)⟩)
171170fveq2d 6348 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑔, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑤⟩, ⟨𝑦, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)) = ((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), ((Id‘𝐷)‘𝑤)⟩))
172 df-ov 6808 . . . . . . . 8 ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)(⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓), ((Id‘𝐷)‘𝑤)⟩)
173171, 172syl6eqr 2804 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘(⟨𝑔, ((Id‘𝐷)‘𝑤)⟩(⟨⟨𝑥, 𝑤⟩, ⟨𝑦, 𝑤⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑧, 𝑤⟩)⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)) = ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)(⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)))
174145, 165, 1733eqtr2rd 2793 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) ∧ 𝑤 ∈ (Base‘𝐷)) → ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)(⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((((𝑦(2nd𝐺)𝑧)‘𝑔)‘𝑤)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑤), ((1st ‘((1st𝐺)‘𝑦))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑧))‘𝑤))(((𝑥(2nd𝐺)𝑦)‘𝑓)‘𝑤)))
175174mpteq2dva 4888 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑤 ∈ (Base‘𝐷) ↦ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)(⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤))) = (𝑤 ∈ (Base‘𝐷) ↦ ((((𝑦(2nd𝐺)𝑧)‘𝑔)‘𝑤)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑤), ((1st ‘((1st𝐺)‘𝑦))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑧))‘𝑤))(((𝑥(2nd𝐺)𝑦)‘𝑓)‘𝑤))))
1762, 9, 26, 107, 113, 119, 126, 139, 133catcocl 16539 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
177 eqid 2752 . . . . . 6 ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
1781, 2, 107, 109, 111, 6, 9, 10, 113, 126, 176, 177curf2 17062 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (𝑤 ∈ (Base‘𝐷) ↦ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)(⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤))))
1791, 2, 107, 109, 111, 6, 9, 10, 113, 119, 139, 141, 23curf2cl 17064 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐺)𝑦)‘𝑓) ∈ (((1st𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st𝐺)‘𝑦)))
1801, 2, 107, 109, 111, 6, 9, 10, 119, 126, 133, 135, 23curf2cl 17064 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐺)𝑧)‘𝑔) ∈ (((1st𝐺)‘𝑦)(𝐷 Nat 𝐸)((1st𝐺)‘𝑧)))
18121, 23, 6, 148, 27, 179, 180fucco 16815 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝑄)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓)) = (𝑤 ∈ (Base‘𝐷) ↦ ((((𝑦(2nd𝐺)𝑧)‘𝑔)‘𝑤)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑤), ((1st ‘((1st𝐺)‘𝑦))‘𝑤)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑧))‘𝑤))(((𝑥(2nd𝐺)𝑦)‘𝑓)‘𝑤))))
182175, 178, 1813eqtr4d 2796 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝑄)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓)))
1832, 22, 9, 24, 8, 25, 26, 27, 3, 31, 40, 46, 62, 106, 182isfuncd 16718 . . 3 (𝜑 → (1st𝐺)(𝐶 Func 𝑄)(2nd𝐺))
184 df-br 4797 . . 3 ((1st𝐺)(𝐶 Func 𝑄)(2nd𝐺) ↔ ⟨(1st𝐺), (2nd𝐺)⟩ ∈ (𝐶 Func 𝑄))
185183, 184sylib 208 . 2 (𝜑 → ⟨(1st𝐺), (2nd𝐺)⟩ ∈ (𝐶 Func 𝑄))
18620, 185eqeltrd 2831 1 (𝜑𝐺 ∈ (𝐶 Func 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1624  wcel 2131  Vcvv 3332  cop 4319   class class class wbr 4796  cmpt 4873   × cxp 5256  ccom 5262  Rel wrel 5263   Fn wfn 6036  wf 6037  cfv 6041  (class class class)co 6805  cmpt2 6807  1st c1st 7323  2nd c2nd 7324  Basecbs 16051  Hom chom 16146  compcco 16147  Catccat 16518  Idccid 16519   Func cfunc 16707   Nat cnat 16794   FuncCat cfuc 16795   ×c cxpc 17001   curryF ccurf 17043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-7 11268  df-8 11269  df-9 11270  df-n0 11477  df-z 11562  df-dec 11678  df-uz 11872  df-fz 12512  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-hom 16160  df-cco 16161  df-cat 16522  df-cid 16523  df-func 16711  df-nat 16796  df-fuc 16797  df-xpc 17005  df-curf 17047
This theorem is referenced by:  uncfcurf  17072  diagcl  17074  curf2ndf  17080  yoncl  17095
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