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Theorem curf1cl 18160
Description: The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-Jan-2017.)
Hypotheses
Ref Expression
curfval.g 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curfval.a 𝐴 = (Base‘𝐶)
curfval.c (𝜑𝐶 ∈ Cat)
curfval.d (𝜑𝐷 ∈ Cat)
curfval.f (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curfval.b 𝐵 = (Base‘𝐷)
curf1.x (𝜑𝑋𝐴)
curf1.k 𝐾 = ((1st𝐺)‘𝑋)
Assertion
Ref Expression
curf1cl (𝜑𝐾 ∈ (𝐷 Func 𝐸))

Proof of Theorem curf1cl
Dummy variables 𝑔 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curfval.g . . . 4 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2 curfval.a . . . 4 𝐴 = (Base‘𝐶)
3 curfval.c . . . 4 (𝜑𝐶 ∈ Cat)
4 curfval.d . . . 4 (𝜑𝐷 ∈ Cat)
5 curfval.f . . . 4 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6 curfval.b . . . 4 𝐵 = (Base‘𝐷)
7 curf1.x . . . 4 (𝜑𝑋𝐴)
8 curf1.k . . . 4 𝐾 = ((1st𝐺)‘𝑋)
9 eqid 2731 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
10 eqid 2731 . . . 4 (Id‘𝐶) = (Id‘𝐶)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 18157 . . 3 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
126fvexi 6889 . . . . . . 7 𝐵 ∈ V
1312mptex 7206 . . . . . 6 (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)) ∈ V
1412, 12mpoex 8045 . . . . . 6 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
1513, 14op1std 7964 . . . . 5 (𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ → (1st𝐾) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)))
1611, 15syl 17 . . . 4 (𝜑 → (1st𝐾) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)))
1713, 14op2ndd 7965 . . . . 5 (𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ → (2nd𝐾) = (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))))
1811, 17syl 17 . . . 4 (𝜑 → (2nd𝐾) = (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))))
1916, 18opeq12d 4871 . . 3 (𝜑 → ⟨(1st𝐾), (2nd𝐾)⟩ = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
2011, 19eqtr4d 2774 . 2 (𝜑𝐾 = ⟨(1st𝐾), (2nd𝐾)⟩)
21 eqid 2731 . . . 4 (Base‘𝐸) = (Base‘𝐸)
22 eqid 2731 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
23 eqid 2731 . . . 4 (Id‘𝐷) = (Id‘𝐷)
24 eqid 2731 . . . 4 (Id‘𝐸) = (Id‘𝐸)
25 eqid 2731 . . . 4 (comp‘𝐷) = (comp‘𝐷)
26 eqid 2731 . . . 4 (comp‘𝐸) = (comp‘𝐸)
27 funcrcl 17792 . . . . . 6 (𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸) → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
285, 27syl 17 . . . . 5 (𝜑 → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
2928simprd 496 . . . 4 (𝜑𝐸 ∈ Cat)
30 eqid 2731 . . . . . . . . 9 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
3130, 2, 6xpcbas 18109 . . . . . . . 8 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
32 relfunc 17791 . . . . . . . . 9 Rel ((𝐶 ×c 𝐷) Func 𝐸)
33 1st2ndbr 8007 . . . . . . . . 9 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
3432, 5, 33sylancr 587 . . . . . . . 8 (𝜑 → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
3531, 21, 34funcf1 17795 . . . . . . 7 (𝜑 → (1st𝐹):(𝐴 × 𝐵)⟶(Base‘𝐸))
3635adantr 481 . . . . . 6 ((𝜑𝑦𝐵) → (1st𝐹):(𝐴 × 𝐵)⟶(Base‘𝐸))
377adantr 481 . . . . . 6 ((𝜑𝑦𝐵) → 𝑋𝐴)
38 simpr 485 . . . . . 6 ((𝜑𝑦𝐵) → 𝑦𝐵)
3936, 37, 38fovcdmd 7559 . . . . 5 ((𝜑𝑦𝐵) → (𝑋(1st𝐹)𝑦) ∈ (Base‘𝐸))
4016, 39fmpt3d 7097 . . . 4 (𝜑 → (1st𝐾):𝐵⟶(Base‘𝐸))
41 eqid 2731 . . . . . 6 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) = (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))
42 ovex 7423 . . . . . . 7 (𝑦(Hom ‘𝐷)𝑧) ∈ V
4342mptex 7206 . . . . . 6 (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V
4441, 43fnmpoi 8035 . . . . 5 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) Fn (𝐵 × 𝐵)
4518fneq1d 6628 . . . . 5 (𝜑 → ((2nd𝐾) Fn (𝐵 × 𝐵) ↔ (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) Fn (𝐵 × 𝐵)))
4644, 45mpbiri 257 . . . 4 (𝜑 → (2nd𝐾) Fn (𝐵 × 𝐵))
4718oveqd 7407 . . . . . 6 (𝜑 → (𝑦(2nd𝐾)𝑧) = (𝑦(𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))𝑧))
4841ovmpt4g 7535 . . . . . . 7 ((𝑦𝐵𝑧𝐵 ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V) → (𝑦(𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))
4943, 48mp3an3 1450 . . . . . 6 ((𝑦𝐵𝑧𝐵) → (𝑦(𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))
5047, 49sylan9eq 2791 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → (𝑦(2nd𝐾)𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))
51 eqid 2731 . . . . . . . 8 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
5234ad2antrr 724 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
537ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑋𝐴)
54 simplrl 775 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦𝐵)
5553, 54opelxpd 5704 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
56 simplrr 776 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧𝐵)
5753, 56opelxpd 5704 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
5831, 51, 22, 52, 55, 57funcf2 17797 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩):(⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩)⟶(((1st𝐹)‘⟨𝑋, 𝑦⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑋, 𝑧⟩)))
59 eqid 2731 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
6030, 31, 59, 9, 51, 55, 57xpchom 18111 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩) = (((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)) × ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩))))
613ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat)
624ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat)
635ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
641, 2, 61, 62, 63, 6, 53, 8, 54curf11 18158 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
65 df-ov 7393 . . . . . . . . . 10 (𝑋(1st𝐹)𝑦) = ((1st𝐹)‘⟨𝑋, 𝑦⟩)
6664, 65eqtr2di 2788 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐹)‘⟨𝑋, 𝑦⟩) = ((1st𝐾)‘𝑦))
671, 2, 61, 62, 63, 6, 53, 8, 56curf11 18158 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐾)‘𝑧) = (𝑋(1st𝐹)𝑧))
68 df-ov 7393 . . . . . . . . . 10 (𝑋(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩)
6967, 68eqtr2di 2788 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐹)‘⟨𝑋, 𝑧⟩) = ((1st𝐾)‘𝑧))
7066, 69oveq12d 7408 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((1st𝐹)‘⟨𝑋, 𝑦⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑋, 𝑧⟩)) = (((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
7160, 70feq23d 6696 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩):(⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩)⟶(((1st𝐹)‘⟨𝑋, 𝑦⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑋, 𝑧⟩)) ↔ (⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩):(((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)) × ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)))⟶(((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧))))
7258, 71mpbid 231 . . . . . 6 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩):(((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)) × ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)))⟶(((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
732, 59, 10, 61, 53catidcl 17605 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
74 op1stg 7966 . . . . . . . . 9 ((𝑋𝐴𝑦𝐵) → (1st ‘⟨𝑋, 𝑦⟩) = 𝑋)
7553, 54, 74syl2anc 584 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘⟨𝑋, 𝑦⟩) = 𝑋)
76 op1stg 7966 . . . . . . . . 9 ((𝑋𝐴𝑧𝐵) → (1st ‘⟨𝑋, 𝑧⟩) = 𝑋)
7753, 56, 76syl2anc 584 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘⟨𝑋, 𝑧⟩) = 𝑋)
7875, 77oveq12d 7408 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)) = (𝑋(Hom ‘𝐶)𝑋))
7973, 78eleqtrrd 2835 . . . . . 6 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑋) ∈ ((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)))
80 simpr 485 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
81 op2ndg 7967 . . . . . . . . 9 ((𝑋𝐴𝑦𝐵) → (2nd ‘⟨𝑋, 𝑦⟩) = 𝑦)
8253, 54, 81syl2anc 584 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘⟨𝑋, 𝑦⟩) = 𝑦)
83 op2ndg 7967 . . . . . . . . 9 ((𝑋𝐴𝑧𝐵) → (2nd ‘⟨𝑋, 𝑧⟩) = 𝑧)
8453, 56, 83syl2anc 584 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘⟨𝑋, 𝑧⟩) = 𝑧)
8582, 84oveq12d 7408 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)) = (𝑦(Hom ‘𝐷)𝑧))
8680, 85eleqtrrd 2835 . . . . . 6 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)))
8772, 79, 86fovcdmd 7559 . . . . 5 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) ∈ (((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
8850, 87fmpt3d 7097 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → (𝑦(2nd𝐾)𝑧):(𝑦(Hom ‘𝐷)𝑧)⟶(((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
893adantr 481 . . . . . . . . 9 ((𝜑𝑦𝐵) → 𝐶 ∈ Cat)
904adantr 481 . . . . . . . . 9 ((𝜑𝑦𝐵) → 𝐷 ∈ Cat)
91 eqid 2731 . . . . . . . . 9 (Id‘(𝐶 ×c 𝐷)) = (Id‘(𝐶 ×c 𝐷))
9230, 89, 90, 2, 6, 10, 23, 91, 37, 38xpcid 18120 . . . . . . . 8 ((𝜑𝑦𝐵) → ((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩) = ⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩)
9392fveq2d 6879 . . . . . . 7 ((𝜑𝑦𝐵) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩))
94 df-ov 7393 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩)
9593, 94eqtr4di 2789 . . . . . 6 ((𝜑𝑦𝐵) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)))
9634adantr 481 . . . . . . 7 ((𝜑𝑦𝐵) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
97 opelxpi 5703 . . . . . . . 8 ((𝑋𝐴𝑦𝐵) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
987, 97sylan 580 . . . . . . 7 ((𝜑𝑦𝐵) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
9931, 91, 24, 96, 98funcid 17799 . . . . . 6 ((𝜑𝑦𝐵) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑋, 𝑦⟩)))
10095, 99eqtr3d 2773 . . . . 5 ((𝜑𝑦𝐵) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑋, 𝑦⟩)))
1015adantr 481 . . . . . 6 ((𝜑𝑦𝐵) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
1026, 9, 23, 90, 38catidcl 17605 . . . . . 6 ((𝜑𝑦𝐵) → ((Id‘𝐷)‘𝑦) ∈ (𝑦(Hom ‘𝐷)𝑦))
1031, 2, 89, 90, 101, 6, 37, 8, 38, 9, 10, 38, 102curf12 18159 . . . . 5 ((𝜑𝑦𝐵) → ((𝑦(2nd𝐾)𝑦)‘((Id‘𝐷)‘𝑦)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)))
1041, 2, 89, 90, 101, 6, 37, 8, 38curf11 18158 . . . . . . 7 ((𝜑𝑦𝐵) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
105104, 65eqtrdi 2787 . . . . . 6 ((𝜑𝑦𝐵) → ((1st𝐾)‘𝑦) = ((1st𝐹)‘⟨𝑋, 𝑦⟩))
106105fveq2d 6879 . . . . 5 ((𝜑𝑦𝐵) → ((Id‘𝐸)‘((1st𝐾)‘𝑦)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑋, 𝑦⟩)))
107100, 103, 1063eqtr4d 2781 . . . 4 ((𝜑𝑦𝐵) → ((𝑦(2nd𝐾)𝑦)‘((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st𝐾)‘𝑦)))
10873ad2ant1 1133 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋𝐴)
109 simp21 1206 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑦𝐵)
110 simp22 1207 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧𝐵)
111 eqid 2731 . . . . . . . . . 10 (comp‘𝐶) = (comp‘𝐶)
112 eqid 2731 . . . . . . . . . 10 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
113 simp23 1208 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤𝐵)
11433ad2ant1 1133 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat)
1152, 59, 10, 114, 108catidcl 17605 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
116 simp3l 1201 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
117 simp3r 1202 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ∈ (𝑧(Hom ‘𝐷)𝑤))
11830, 2, 6, 59, 9, 108, 109, 108, 110, 111, 25, 112, 108, 113, 115, 116, 115, 117xpcco2 18118 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((Id‘𝐶)‘𝑋), ⟩(⟨⟨𝑋, 𝑦⟩, ⟨𝑋, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑔⟩) = ⟨(((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑋)), ((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩)
1192, 59, 10, 114, 108, 111, 108, 115catlid 17606 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑋)) = ((Id‘𝐶)‘𝑋))
120119opeq1d 4869 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨(((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑋)), ((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩ = ⟨((Id‘𝐶)‘𝑋), ((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩)
121118, 120eqtrd 2771 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((Id‘𝐶)‘𝑋), ⟩(⟨⟨𝑋, 𝑦⟩, ⟨𝑋, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑔⟩) = ⟨((Id‘𝐶)‘𝑋), ((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩)
122121fveq2d 6879 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘(⟨((Id‘𝐶)‘𝑋), ⟩(⟨⟨𝑋, 𝑦⟩, ⟨𝑋, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑔⟩)) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩))
123 df-ov 7393 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑤⟩)((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)⟩)
124122, 123eqtr4di 2789 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘(⟨((Id‘𝐶)‘𝑋), ⟩(⟨⟨𝑋, 𝑦⟩, ⟨𝑋, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑔⟩)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑤⟩)((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)))
125343ad2ant1 1133 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
126108, 109opelxpd 5704 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
127108, 110opelxpd 5704 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
128108, 113opelxpd 5704 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
129115, 116opelxpd 5704 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑔⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑦(Hom ‘𝐷)𝑧)))
13030, 2, 6, 59, 9, 108, 109, 108, 110, 51xpchom2 18117 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑦(Hom ‘𝐷)𝑧)))
131129, 130eleqtrrd 2835 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑔⟩ ∈ (⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩))
132115, 117opelxpd 5704 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), ⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
13330, 2, 6, 59, 9, 108, 110, 108, 113, 51xpchom2 18117 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
134132, 133eleqtrrd 2835 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), ⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩))
13531, 51, 112, 26, 125, 126, 127, 128, 131, 134funcco 17800 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘(⟨((Id‘𝐶)‘𝑋), ⟩(⟨⟨𝑋, 𝑦⟩, ⟨𝑋, 𝑧⟩⟩(comp‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩)⟨((Id‘𝐶)‘𝑋), 𝑔⟩)) = (((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ⟩)(⟨((1st𝐹)‘⟨𝑋, 𝑦⟩), ((1st𝐹)‘⟨𝑋, 𝑧⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑋, 𝑤⟩))((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩)))
136124, 135eqtr3d 2773 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑤⟩)((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)) = (((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ⟩)(⟨((1st𝐹)‘⟨𝑋, 𝑦⟩), ((1st𝐹)‘⟨𝑋, 𝑧⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑋, 𝑤⟩))((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩)))
13743ad2ant1 1133 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat)
13853ad2ant1 1133 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
1396, 9, 25, 137, 109, 110, 113, 116, 117catcocl 17608 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐷)𝑤))
1401, 2, 114, 137, 138, 6, 108, 8, 109, 9, 10, 113, 139curf12 18159 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd𝐾)𝑤)‘((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑤⟩)((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)))
1411, 2, 114, 137, 138, 6, 108, 8, 109curf11 18158 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
142141, 65eqtrdi 2787 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑦) = ((1st𝐹)‘⟨𝑋, 𝑦⟩))
1431, 2, 114, 137, 138, 6, 108, 8, 110curf11 18158 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑧) = (𝑋(1st𝐹)𝑧))
144143, 68eqtrdi 2787 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
145142, 144opeq12d 4871 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1st𝐾)‘𝑦), ((1st𝐾)‘𝑧)⟩ = ⟨((1st𝐹)‘⟨𝑋, 𝑦⟩), ((1st𝐹)‘⟨𝑋, 𝑧⟩)⟩)
1461, 2, 114, 137, 138, 6, 108, 8, 113curf11 18158 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑤) = (𝑋(1st𝐹)𝑤))
147 df-ov 7393 . . . . . . . 8 (𝑋(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩)
148146, 147eqtrdi 2787 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩))
149145, 148oveq12d 7408 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (⟨((1st𝐾)‘𝑦), ((1st𝐾)‘𝑧)⟩(comp‘𝐸)((1st𝐾)‘𝑤)) = (⟨((1st𝐹)‘⟨𝑋, 𝑦⟩), ((1st𝐹)‘⟨𝑋, 𝑧⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑋, 𝑤⟩)))
1501, 2, 114, 137, 138, 6, 108, 8, 110, 9, 10, 113, 117curf12 18159 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd𝐾)𝑤)‘) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)))
151 df-ov 7393 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ⟩)
152150, 151eqtrdi 2787 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd𝐾)𝑤)‘) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ⟩))
1531, 2, 114, 137, 138, 6, 108, 8, 109, 9, 10, 110, 116curf12 18159 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd𝐾)𝑧)‘𝑔) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))
154 df-ov 7393 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩)
155153, 154eqtrdi 2787 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd𝐾)𝑧)‘𝑔) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩))
156149, 152, 155oveq123d 7411 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((𝑧(2nd𝐾)𝑤)‘)(⟨((1st𝐾)‘𝑦), ((1st𝐾)‘𝑧)⟩(comp‘𝐸)((1st𝐾)‘𝑤))((𝑦(2nd𝐾)𝑧)‘𝑔)) = (((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ⟩)(⟨((1st𝐹)‘⟨𝑋, 𝑦⟩), ((1st𝐹)‘⟨𝑋, 𝑧⟩)⟩(comp‘𝐸)((1st𝐹)‘⟨𝑋, 𝑤⟩))((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩)))
157136, 140, 1563eqtr4d 2781 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd𝐾)𝑤)‘((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)) = (((𝑧(2nd𝐾)𝑤)‘)(⟨((1st𝐾)‘𝑦), ((1st𝐾)‘𝑧)⟩(comp‘𝐸)((1st𝐾)‘𝑤))((𝑦(2nd𝐾)𝑧)‘𝑔)))
1586, 21, 9, 22, 23, 24, 25, 26, 4, 29, 40, 46, 88, 107, 157isfuncd 17794 . . 3 (𝜑 → (1st𝐾)(𝐷 Func 𝐸)(2nd𝐾))
159 df-br 5139 . . 3 ((1st𝐾)(𝐷 Func 𝐸)(2nd𝐾) ↔ ⟨(1st𝐾), (2nd𝐾)⟩ ∈ (𝐷 Func 𝐸))
160158, 159sylib 217 . 2 (𝜑 → ⟨(1st𝐾), (2nd𝐾)⟩ ∈ (𝐷 Func 𝐸))
16120, 160eqeltrd 2832 1 (𝜑𝐾 ∈ (𝐷 Func 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3470  cop 4625   class class class wbr 5138  cmpt 5221   × cxp 5664  Rel wrel 5671   Fn wfn 6524  wf 6525  cfv 6529  (class class class)co 7390  cmpo 7392  1st c1st 7952  2nd c2nd 7953  Basecbs 17123  Hom chom 17187  compcco 17188  Catccat 17587  Idccid 17588   Func cfunc 17783   ×c cxpc 18099   curryF ccurf 18142
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7705  ax-cnex 11145  ax-resscn 11146  ax-1cn 11147  ax-icn 11148  ax-addcl 11149  ax-addrcl 11150  ax-mulcl 11151  ax-mulrcl 11152  ax-mulcom 11153  ax-addass 11154  ax-mulass 11155  ax-distr 11156  ax-i2m1 11157  ax-1ne0 11158  ax-1rid 11159  ax-rnegex 11160  ax-rrecex 11161  ax-cnre 11162  ax-pre-lttri 11163  ax-pre-lttrn 11164  ax-pre-ltadd 11165  ax-pre-mulgt0 11166
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-tp 4624  df-op 4626  df-uni 4899  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6286  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-f1 6534  df-fo 6535  df-f1o 6536  df-fv 6537  df-riota 7346  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7836  df-1st 7954  df-2nd 7955  df-frecs 8245  df-wrecs 8276  df-recs 8350  df-rdg 8389  df-1o 8445  df-er 8683  df-map 8802  df-ixp 8872  df-en 8920  df-dom 8921  df-sdom 8922  df-fin 8923  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11425  df-neg 11426  df-nn 12192  df-2 12254  df-3 12255  df-4 12256  df-5 12257  df-6 12258  df-7 12259  df-8 12260  df-9 12261  df-n0 12452  df-z 12538  df-dec 12657  df-uz 12802  df-fz 13464  df-struct 17059  df-slot 17094  df-ndx 17106  df-base 17124  df-hom 17200  df-cco 17201  df-cat 17591  df-cid 17592  df-func 17787  df-xpc 18103  df-curf 18146
This theorem is referenced by:  curf2cl  18163  curfcl  18164
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