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Theorem curfcl 18182
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 2733 . . . 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 2733 . . . 4 (Base‘𝐷) = (Base‘𝐷)
7 eqid 2733 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
8 eqid 2733 . . . 4 (Id‘𝐶) = (Id‘𝐶)
9 eqid 2733 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
10 eqid 2733 . . . 4 (Id‘𝐷) = (Id‘𝐷)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curfval 18173 . . 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 6902 . . . . . . 7 (Base‘𝐶) ∈ V
1312mptex 7222 . . . . . 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, 12mpoex 8063 . . . . . 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 7982 . . . . 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 7983 . . . . 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 4881 . . 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 2776 . 2 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
21 curfcl.q . . . . 5 𝑄 = (𝐷 FuncCat 𝐸)
2221fucbas 17909 . . . 4 (𝐷 Func 𝐸) = (Base‘𝑄)
23 eqid 2733 . . . . 5 (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
2421, 23fuchom 17910 . . . 4 (𝐷 Nat 𝐸) = (Hom ‘𝑄)
25 eqid 2733 . . . 4 (Id‘𝑄) = (Id‘𝑄)
26 eqid 2733 . . . 4 (comp‘𝐶) = (comp‘𝐶)
27 eqid 2733 . . . 4 (comp‘𝑄) = (comp‘𝑄)
28 funcrcl 17810 . . . . . . 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 497 . . . . 5 (𝜑𝐸 ∈ Cat)
3121, 4, 30fuccat 17920 . . . 4 (𝜑𝑄 ∈ Cat)
32 opex 5464 . . . . . 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 482 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
354adantr 482 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
365adantr 482 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
37 simpr 486 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
38 eqid 2733 . . . . . 6 ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑥)
391, 2, 34, 35, 36, 6, 37, 38curf1cl 18178 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
4033, 16, 39fmpt2d 7120 . . . 4 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(𝐷 Func 𝐸))
41 eqid 2733 . . . . . 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 7439 . . . . . . 7 (𝑥(Hom ‘𝐶)𝑦) ∈ V
4342mptex 7222 . . . . . 6 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))) ∈ V
4441, 43fnmpoi 8053 . . . . 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 6640 . . . . 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 258 . . . 4 (𝜑 → (2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)))
47 fvex 6902 . . . . . . 7 (Base‘𝐷) ∈ V
4847mptex 7222 . . . . . 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 7423 . . . . . 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 7552 . . . . . . 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 1451 . . . . . 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 2793 . . . . 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 725 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐶 ∈ Cat)
554ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
565ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
57 simplrl 776 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
58 simplrr 777 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
59 simpr 486 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
60 eqid 2733 . . . . . 6 ((𝑥(2nd𝐺)𝑦)‘𝑔) = ((𝑥(2nd𝐺)𝑦)‘𝑔)
611, 2, 54, 55, 56, 6, 9, 10, 57, 58, 59, 60, 23curf2cl 18181 . . . . 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 7120 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st𝐺)‘𝑦)))
63 eqid 2733 . . . . . . . . . 10 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
6463, 2, 6xpcbas 18127 . . . . . . . . 9 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×c 𝐷))
65 eqid 2733 . . . . . . . . 9 (Id‘(𝐶 ×c 𝐷)) = (Id‘(𝐶 ×c 𝐷))
66 eqid 2733 . . . . . . . . 9 (Id‘𝐸) = (Id‘𝐸)
67 relfunc 17809 . . . . . . . . . . 11 Rel ((𝐶 ×c 𝐷) Func 𝐸)
68 1st2ndbr 8025 . . . . . . . . . . 11 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
6967, 5, 68sylancr 588 . . . . . . . . . 10 (𝜑 → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
7069ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
71 opelxpi 5713 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
7271adantll 713 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
7364, 65, 66, 70, 72funcid 17817 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑥, 𝑦⟩)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑥, 𝑦⟩)))
743ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat)
754ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
7637adantr 482 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
77 simpr 486 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐷))
7863, 74, 75, 2, 6, 8, 10, 65, 76, 77xpcid 18138 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((Id‘(𝐶 ×c 𝐷))‘⟨𝑥, 𝑦⟩) = ⟨((Id‘𝐶)‘𝑥), ((Id‘𝐷)‘𝑦)⟩)
7978fveq2d 6893 . . . . . . . . 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 7409 . . . . . . . . 9 (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑥), ((Id‘𝐷)‘𝑦)⟩)
8179, 80eqtr4di 2791 . . . . . . . 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 725 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
831, 2, 74, 75, 82, 6, 76, 38, 77curf11 18176 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑦) = (𝑥(1st𝐹)𝑦))
84 df-ov 7409 . . . . . . . . . 10 (𝑥(1st𝐹)𝑦) = ((1st𝐹)‘⟨𝑥, 𝑦⟩)
8583, 84eqtr2di 2790 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st𝐹)‘⟨𝑥, 𝑦⟩) = ((1st ‘((1st𝐺)‘𝑥))‘𝑦))
8685fveq2d 6893 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((Id‘𝐸)‘((1st𝐹)‘⟨𝑥, 𝑦⟩)) = ((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑦)))
8773, 81, 863eqtr3d 2781 . . . . . . 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 5248 . . . . . 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 482 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
90 eqid 2733 . . . . . . . . . 10 (Base‘𝐸) = (Base‘𝐸)
9190, 66cidfn 17620 . . . . . . . . 9 (𝐸 ∈ Cat → (Id‘𝐸) Fn (Base‘𝐸))
9289, 91syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (Id‘𝐸) Fn (Base‘𝐸))
93 dffn2 6717 . . . . . . . 8 ((Id‘𝐸) Fn (Base‘𝐸) ↔ (Id‘𝐸):(Base‘𝐸)⟶V)
9492, 93sylib 217 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (Id‘𝐸):(Base‘𝐸)⟶V)
95 relfunc 17809 . . . . . . . . 9 Rel (𝐷 Func 𝐸)
96 1st2ndbr 8025 . . . . . . . . 9 ((Rel (𝐷 Func 𝐸) ∧ ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
9795, 39, 96sylancr 588 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
986, 90, 97funcf1 17813 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
99 fcompt 7128 . . . . . . 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 585 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐸) ∘ (1st ‘((1st𝐺)‘𝑥))) = (𝑦 ∈ (Base‘𝐷) ↦ ((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑦))))
10188, 100eqtr4d 2776 . . . . 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 17623 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
103 eqid 2733 . . . . . 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 18179 . . . . 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 17921 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑄)‘((1st𝐺)‘𝑥)) = ((Id‘𝐸) ∘ (1st ‘((1st𝐺)‘𝑥))))
106101, 104, 1053eqtr4d 2783 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝑄)‘((1st𝐺)‘𝑥)))
10733ad2ant1 1134 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
108107adantr 482 . . . . . . . . . . . 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 1134 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
110109adantr 482 . . . . . . . . . . . 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 1134 . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 1207 . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 486 . . . . . . . . . . . 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 18176 . . . . . . . . . . 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 7409 . . . . . . . . . . 11 (𝑥(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑥, 𝑤⟩)
118116, 117eqtrdi 2789 . . . . . . . . . 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 1208 . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 2733 . . . . . . . . . . . 12 ((1st𝐺)‘𝑦) = ((1st𝐺)‘𝑦)
1221, 2, 108, 110, 112, 6, 120, 121, 115curf11 18176 . . . . . . . . . . 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 7409 . . . . . . . . . . 11 (𝑦(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑦, 𝑤⟩)
124122, 123eqtrdi 2789 . . . . . . . . . 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 4881 . . . . . . . . 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 1209 . . . . . . . . . . . 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 482 . . . . . . . . . . 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 2733 . . . . . . . . . . 11 ((1st𝐺)‘𝑧) = ((1st𝐺)‘𝑧)
1291, 2, 108, 110, 112, 6, 127, 128, 115curf11 18176 . . . . . . . . . 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 7409 . . . . . . . . . 10 (𝑧(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑧, 𝑤⟩)
131129, 130eqtrdi 2789 . . . . . . . . 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 7424 . . . . . . . 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 1203 . . . . . . . . . . 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 482 . . . . . . . . . 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 2733 . . . . . . . . . 10 ((𝑦(2nd𝐺)𝑧)‘𝑔) = ((𝑦(2nd𝐺)𝑧)‘𝑔)
1361, 2, 108, 110, 112, 6, 9, 10, 120, 127, 134, 135, 115curf2val 18180 . . . . . . . . 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 7409 . . . . . . . . 9 (𝑔(⟨𝑦, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑦, 𝑤⟩(2nd𝐹)⟨𝑧, 𝑤⟩)‘⟨𝑔, ((Id‘𝐷)‘𝑤)⟩)
138136, 137eqtrdi 2789 . . . . . . . 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 1202 . . . . . . . . . . 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 482 . . . . . . . . . 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 2733 . . . . . . . . . 10 ((𝑥(2nd𝐺)𝑦)‘𝑓) = ((𝑥(2nd𝐺)𝑦)‘𝑓)
1421, 2, 108, 110, 112, 6, 9, 10, 114, 120, 140, 141, 115curf2val 18180 . . . . . . . . 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 7409 . . . . . . . . 9 (𝑓(⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑦, 𝑤⟩)((Id‘𝐷)‘𝑤)) = ((⟨𝑥, 𝑤⟩(2nd𝐹)⟨𝑦, 𝑤⟩)‘⟨𝑓, ((Id‘𝐷)‘𝑤)⟩)
144142, 143eqtrdi 2789 . . . . . . . 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 7427 . . . . . . 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 2733 . . . . . . . 8 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
147 eqid 2733 . . . . . . . 8 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
148 eqid 2733 . . . . . . . 8 (comp‘𝐸) = (comp‘𝐸)
14967, 112, 68sylancr 588 . . . . . . . 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 5713 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
151113, 150sylan 581 . . . . . . . 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 5713 . . . . . . . . 9 ((𝑦 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑦, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
153119, 152sylan 581 . . . . . . . 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 5713 . . . . . . . . 9 ((𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → ⟨𝑧, 𝑤⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
155126, 154sylan 581 . . . . . . . 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 17623 . . . . . . . . . 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))
157140, 156opelxpd 5714 . . . . . . . . 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)))
15863, 2, 6, 9, 7, 114, 115, 120, 115, 146xpchom2 18135 . . . . . . . . 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)))
159157, 158eleqtrrd 2837 . . . . . . . 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⟩))
160134, 156opelxpd 5714 . . . . . . . . 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)))
16163, 2, 6, 9, 7, 120, 115, 127, 115, 146xpchom2 18135 . . . . . . . . 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)))
162160, 161eleqtrrd 2837 . . . . . . . 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⟩))
16364, 146, 147, 148, 149, 151, 153, 155, 159, 162funcco 17818 . . . . . . 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)⟩)))
164 eqid 2733 . . . . . . . . . . 11 (comp‘𝐷) = (comp‘𝐷)
16563, 2, 6, 9, 7, 114, 115, 120, 115, 26, 164, 147, 127, 115, 140, 156, 134, 156xpcco2 18136 . . . . . . . . . 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))⟩)
1666, 7, 10, 110, 115, 164, 115, 156catlid 17624 . . . . . . . . . . 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))
167166opeq2d 4880 . . . . . . . . . 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)⟩)
168165, 167eqtrd 2773 . . . . . . . . 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)⟩)
169168fveq2d 6893 . . . . . . . 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)⟩))
170 df-ov 7409 . . . . . . . 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)⟩)
171169, 170eqtr4di 2791 . . . . . . 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)))
172145, 163, 1713eqtr2rd 2780 . . . . . 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)))
173172mpteq2dva 5248 . . . . 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))))
1742, 9, 26, 107, 113, 119, 126, 139, 133catcocl 17626 . . . . . 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))
175 eqid 2733 . . . . . 6 ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
1761, 2, 107, 109, 111, 6, 9, 10, 113, 126, 174, 175curf2 18179 . . . . 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))))
1771, 2, 107, 109, 111, 6, 9, 10, 113, 119, 139, 141, 23curf2cl 18181 . . . . . 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)))
1781, 2, 107, 109, 111, 6, 9, 10, 119, 126, 133, 135, 23curf2cl 18181 . . . . . 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)))
17921, 23, 6, 148, 27, 177, 178fucco 17912 . . . . 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))))
180173, 176, 1793eqtr4d 2783 . . . 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)))
1812, 22, 9, 24, 8, 25, 26, 27, 3, 31, 40, 46, 62, 106, 180isfuncd 17812 . . 3 (𝜑 → (1st𝐺)(𝐶 Func 𝑄)(2nd𝐺))
182 df-br 5149 . . 3 ((1st𝐺)(𝐶 Func 𝑄)(2nd𝐺) ↔ ⟨(1st𝐺), (2nd𝐺)⟩ ∈ (𝐶 Func 𝑄))
183181, 182sylib 217 . 2 (𝜑 → ⟨(1st𝐺), (2nd𝐺)⟩ ∈ (𝐶 Func 𝑄))
18420, 183eqeltrd 2834 1 (𝜑𝐺 ∈ (𝐶 Func 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  cop 4634   class class class wbr 5148  cmpt 5231   × cxp 5674  ccom 5680  Rel wrel 5681   Fn wfn 6536  wf 6537  cfv 6541  (class class class)co 7406  cmpo 7408  1st c1st 7970  2nd c2nd 7971  Basecbs 17141  Hom chom 17205  compcco 17206  Catccat 17605  Idccid 17606   Func cfunc 17801   Nat cnat 17889   FuncCat cfuc 17890   ×c cxpc 18117   curryF ccurf 18160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-struct 17077  df-slot 17112  df-ndx 17124  df-base 17142  df-hom 17218  df-cco 17219  df-cat 17609  df-cid 17610  df-func 17805  df-nat 17891  df-fuc 17892  df-xpc 18121  df-curf 18164
This theorem is referenced by:  uncfcurf  18189  diagcl  18191  curf2ndf  18197  yoncl  18212
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