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Theorem curf1cl 17466
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 2818 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
10 eqid 2818 . . . 4 (Id‘𝐶) = (Id‘𝐶)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 17463 . . 3 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
126fvexi 6677 . . . . . . 7 𝐵 ∈ V
1312mptex 6977 . . . . . 6 (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)) ∈ V
1412, 12mpoex 7766 . . . . . 6 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
1513, 14op1std 7688 . . . . 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 7689 . . . . 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 4803 . . 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 2856 . 2 (𝜑𝐾 = ⟨(1st𝐾), (2nd𝐾)⟩)
21 eqid 2818 . . . 4 (Base‘𝐸) = (Base‘𝐸)
22 eqid 2818 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
23 eqid 2818 . . . 4 (Id‘𝐷) = (Id‘𝐷)
24 eqid 2818 . . . 4 (Id‘𝐸) = (Id‘𝐸)
25 eqid 2818 . . . 4 (comp‘𝐷) = (comp‘𝐷)
26 eqid 2818 . . . 4 (comp‘𝐸) = (comp‘𝐸)
27 funcrcl 17121 . . . . . 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 2818 . . . . . . . . 9 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
3130, 2, 6xpcbas 17416 . . . . . . . 8 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
32 relfunc 17120 . . . . . . . . 9 Rel ((𝐶 ×c 𝐷) Func 𝐸)
33 1st2ndbr 7730 . . . . . . . . 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 17124 . . . . . . 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, 38fovrnd 7309 . . . . 5 ((𝜑𝑦𝐵) → (𝑋(1st𝐹)𝑦) ∈ (Base‘𝐸))
4016, 39fmpt3d 6872 . . . 4 (𝜑 → (1st𝐾):𝐵⟶(Base‘𝐸))
41 eqid 2818 . . . . . 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 7178 . . . . . . 7 (𝑦(Hom ‘𝐷)𝑧) ∈ V
4342mptex 6977 . . . . . 6 (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V
4441, 43fnmpoi 7757 . . . . 5 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) Fn (𝐵 × 𝐵)
4518fneq1d 6439 . . . . 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 259 . . . 4 (𝜑 → (2nd𝐾) Fn (𝐵 × 𝐵))
4718oveqd 7162 . . . . . 6 (𝜑 → (𝑦(2nd𝐾)𝑧) = (𝑦(𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))𝑧))
4841ovmpt4g 7286 . . . . . . 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 1441 . . . . . 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 2873 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → (𝑦(2nd𝐾)𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))
51 eqid 2818 . . . . . . . 8 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
5234ad2antrr 722 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
537ad2antrr 722 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑋𝐴)
54 simplrl 773 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦𝐵)
5553, 54opelxpd 5586 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
56 simplrr 774 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧𝐵)
5753, 56opelxpd 5586 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
5831, 51, 22, 52, 55, 57funcf2 17126 . . . . . . 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 2818 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
6030, 31, 59, 9, 51, 55, 57xpchom 17418 . . . . . . . 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 722 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat)
624ad2antrr 722 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat)
635ad2antrr 722 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
641, 2, 61, 62, 63, 6, 53, 8, 54curf11 17464 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
65 df-ov 7148 . . . . . . . . . 10 (𝑋(1st𝐹)𝑦) = ((1st𝐹)‘⟨𝑋, 𝑦⟩)
6664, 65syl6req 2870 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐹)‘⟨𝑋, 𝑦⟩) = ((1st𝐾)‘𝑦))
671, 2, 61, 62, 63, 6, 53, 8, 56curf11 17464 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐾)‘𝑧) = (𝑋(1st𝐹)𝑧))
68 df-ov 7148 . . . . . . . . . 10 (𝑋(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩)
6967, 68syl6req 2870 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐹)‘⟨𝑋, 𝑧⟩) = ((1st𝐾)‘𝑧))
7066, 69oveq12d 7163 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((1st𝐹)‘⟨𝑋, 𝑦⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑋, 𝑧⟩)) = (((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
7160, 70feq23d 6502 . . . . . . 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 233 . . . . . 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 16941 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
74 op1stg 7690 . . . . . . . . 9 ((𝑋𝐴𝑦𝐵) → (1st ‘⟨𝑋, 𝑦⟩) = 𝑋)
7553, 54, 74syl2anc 584 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘⟨𝑋, 𝑦⟩) = 𝑋)
76 op1stg 7690 . . . . . . . . 9 ((𝑋𝐴𝑧𝐵) → (1st ‘⟨𝑋, 𝑧⟩) = 𝑋)
7753, 56, 76syl2anc 584 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘⟨𝑋, 𝑧⟩) = 𝑋)
7875, 77oveq12d 7163 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)) = (𝑋(Hom ‘𝐶)𝑋))
7973, 78eleqtrrd 2913 . . . . . 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 7691 . . . . . . . . 9 ((𝑋𝐴𝑦𝐵) → (2nd ‘⟨𝑋, 𝑦⟩) = 𝑦)
8253, 54, 81syl2anc 584 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘⟨𝑋, 𝑦⟩) = 𝑦)
83 op2ndg 7691 . . . . . . . . 9 ((𝑋𝐴𝑧𝐵) → (2nd ‘⟨𝑋, 𝑧⟩) = 𝑧)
8453, 56, 83syl2anc 584 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘⟨𝑋, 𝑧⟩) = 𝑧)
8582, 84oveq12d 7163 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)) = (𝑦(Hom ‘𝐷)𝑧))
8680, 85eleqtrrd 2913 . . . . . 6 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)))
8772, 79, 86fovrnd 7309 . . . . 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 6872 . . . 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 2818 . . . . . . . . 9 (Id‘(𝐶 ×c 𝐷)) = (Id‘(𝐶 ×c 𝐷))
9230, 89, 90, 2, 6, 10, 23, 91, 37, 38xpcid 17427 . . . . . . . 8 ((𝜑𝑦𝐵) → ((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩) = ⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩)
9392fveq2d 6667 . . . . . . 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 7148 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩)
9593, 94syl6eqr 2871 . . . . . 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 5585 . . . . . . . 8 ((𝑋𝐴𝑦𝐵) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
987, 97sylan 580 . . . . . . 7 ((𝜑𝑦𝐵) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
9931, 91, 24, 96, 98funcid 17128 . . . . . 6 ((𝜑𝑦𝐵) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑋, 𝑦⟩)))
10095, 99eqtr3d 2855 . . . . 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 16941 . . . . . 6 ((𝜑𝑦𝐵) → ((Id‘𝐷)‘𝑦) ∈ (𝑦(Hom ‘𝐷)𝑦))
1031, 2, 89, 90, 101, 6, 37, 8, 38, 9, 10, 38, 102curf12 17465 . . . . 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 17464 . . . . . . 7 ((𝜑𝑦𝐵) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
105104, 65syl6eq 2869 . . . . . 6 ((𝜑𝑦𝐵) → ((1st𝐾)‘𝑦) = ((1st𝐹)‘⟨𝑋, 𝑦⟩))
106105fveq2d 6667 . . . . 5 ((𝜑𝑦𝐵) → ((Id‘𝐸)‘((1st𝐾)‘𝑦)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑋, 𝑦⟩)))
107100, 103, 1063eqtr4d 2863 . . . 4 ((𝜑𝑦𝐵) → ((𝑦(2nd𝐾)𝑦)‘((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st𝐾)‘𝑦)))
10873ad2ant1 1125 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋𝐴)
109 simp21 1198 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑦𝐵)
110 simp22 1199 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧𝐵)
111 eqid 2818 . . . . . . . . . 10 (comp‘𝐶) = (comp‘𝐶)
112 eqid 2818 . . . . . . . . . 10 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
113 simp23 1200 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤𝐵)
11433ad2ant1 1125 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat)
1152, 59, 10, 114, 108catidcl 16941 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
116 simp3l 1193 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
117 simp3r 1194 . . . . . . . . . 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 17425 . . . . . . . . 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 16942 . . . . . . . . . 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 4801 . . . . . . . . 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 2853 . . . . . . . 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 6667 . . . . . . 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 7148 . . . . . . 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, 123syl6eqr 2871 . . . . . 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 1125 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
126108, 109opelxpd 5586 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
127108, 110opelxpd 5586 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
128108, 113opelxpd 5586 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
129115, 116opelxpd 5586 . . . . . . . 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 17424 . . . . . . . 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 2913 . . . . . . 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 5586 . . . . . . . 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 17424 . . . . . . . 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 2913 . . . . . . 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 17129 . . . . . 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 2855 . . . . 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 1125 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat)
13853ad2ant1 1125 . . . . . 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 16944 . . . . . 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 17465 . . . . 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 17464 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
142141, 65syl6eq 2869 . . . . . . . 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 17464 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑧) = (𝑋(1st𝐹)𝑧))
144143, 68syl6eq 2869 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
145142, 144opeq12d 4803 . . . . . . 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 17464 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑤) = (𝑋(1st𝐹)𝑤))
147 df-ov 7148 . . . . . . . 8 (𝑋(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩)
148146, 147syl6eq 2869 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩))
149145, 148oveq12d 7163 . . . . . 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 17465 . . . . . . 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 7148 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ⟩)
152150, 151syl6eq 2869 . . . . . 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 17465 . . . . . . 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 7148 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩)
155153, 154syl6eq 2869 . . . . . 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 7166 . . . . 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 2863 . . . 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 17123 . . 3 (𝜑 → (1st𝐾)(𝐷 Func 𝐸)(2nd𝐾))
159 df-br 5058 . . 3 ((1st𝐾)(𝐷 Func 𝐸)(2nd𝐾) ↔ ⟨(1st𝐾), (2nd𝐾)⟩ ∈ (𝐷 Func 𝐸))
160158, 159sylib 219 . 2 (𝜑 → ⟨(1st𝐾), (2nd𝐾)⟩ ∈ (𝐷 Func 𝐸))
16120, 160eqeltrd 2910 1 (𝜑𝐾 ∈ (𝐷 Func 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  Vcvv 3492  cop 4563   class class class wbr 5057  cmpt 5137   × cxp 5546  Rel wrel 5553   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  Basecbs 16471  Hom chom 16564  compcco 16565  Catccat 16923  Idccid 16924   Func cfunc 17112   ×c cxpc 17406   curryF ccurf 17448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-hom 16577  df-cco 16578  df-cat 16927  df-cid 16928  df-func 17116  df-xpc 17410  df-curf 17452
This theorem is referenced by:  curf2cl  17469  curfcl  17470
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