MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  curf1cl Structured version   Visualization version   GIF version

Theorem curf1cl 17134
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 2765 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
10 eqid 2765 . . . 4 (Id‘𝐶) = (Id‘𝐶)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 17131 . . 3 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
126fvexi 6389 . . . . . . 7 𝐵 ∈ V
1312mptex 6679 . . . . . 6 (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)) ∈ V
1412, 12mpt2ex 7448 . . . . . 6 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
1513, 14op1std 7376 . . . . 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 7377 . . . . 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 4567 . . 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 2802 . 2 (𝜑𝐾 = ⟨(1st𝐾), (2nd𝐾)⟩)
21 eqid 2765 . . . 4 (Base‘𝐸) = (Base‘𝐸)
22 eqid 2765 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
23 eqid 2765 . . . 4 (Id‘𝐷) = (Id‘𝐷)
24 eqid 2765 . . . 4 (Id‘𝐸) = (Id‘𝐸)
25 eqid 2765 . . . 4 (comp‘𝐷) = (comp‘𝐷)
26 eqid 2765 . . . 4 (comp‘𝐸) = (comp‘𝐸)
27 funcrcl 16788 . . . . . 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 489 . . . 4 (𝜑𝐸 ∈ Cat)
30 eqid 2765 . . . . . . . . . 10 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
3130, 2, 6xpcbas 17084 . . . . . . . . 9 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
32 relfunc 16787 . . . . . . . . . 10 Rel ((𝐶 ×c 𝐷) Func 𝐸)
33 1st2ndbr 7417 . . . . . . . . . 10 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
3432, 5, 33sylancr 581 . . . . . . . . 9 (𝜑 → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
3531, 21, 34funcf1 16791 . . . . . . . 8 (𝜑 → (1st𝐹):(𝐴 × 𝐵)⟶(Base‘𝐸))
3635adantr 472 . . . . . . 7 ((𝜑𝑦𝐵) → (1st𝐹):(𝐴 × 𝐵)⟶(Base‘𝐸))
377adantr 472 . . . . . . 7 ((𝜑𝑦𝐵) → 𝑋𝐴)
38 simpr 477 . . . . . . 7 ((𝜑𝑦𝐵) → 𝑦𝐵)
3936, 37, 38fovrnd 7004 . . . . . 6 ((𝜑𝑦𝐵) → (𝑋(1st𝐹)𝑦) ∈ (Base‘𝐸))
4039fmpttd 6575 . . . . 5 (𝜑 → (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)):𝐵⟶(Base‘𝐸))
4116feq1d 6208 . . . . 5 (𝜑 → ((1st𝐾):𝐵⟶(Base‘𝐸) ↔ (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)):𝐵⟶(Base‘𝐸)))
4240, 41mpbird 248 . . . 4 (𝜑 → (1st𝐾):𝐵⟶(Base‘𝐸))
43 eqid 2765 . . . . . 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)))
44 ovex 6874 . . . . . . 7 (𝑦(Hom ‘𝐷)𝑧) ∈ V
4544mptex 6679 . . . . . 6 (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V
4643, 45fnmpt2i 7440 . . . . 5 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) Fn (𝐵 × 𝐵)
4718fneq1d 6159 . . . . 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)))
4846, 47mpbiri 249 . . . 4 (𝜑 → (2nd𝐾) Fn (𝐵 × 𝐵))
49 eqid 2765 . . . . . . . . 9 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
5034ad2antrr 717 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
517ad2antrr 717 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑋𝐴)
52 simplrl 795 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦𝐵)
53 opelxpi 5314 . . . . . . . . . 10 ((𝑋𝐴𝑦𝐵) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
5451, 52, 53syl2anc 579 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
55 simplrr 796 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧𝐵)
56 opelxpi 5314 . . . . . . . . . 10 ((𝑋𝐴𝑧𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
5751, 55, 56syl2anc 579 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
5831, 49, 22, 50, 54, 57funcf2 16793 . . . . . . . 8 (((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 2765 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
6030, 31, 59, 9, 49, 54, 57xpchom 17086 . . . . . . . . 9 (((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 717 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat)
624ad2antrr 717 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat)
635ad2antrr 717 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
641, 2, 61, 62, 63, 6, 51, 8, 52curf11 17132 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
65 df-ov 6845 . . . . . . . . . . 11 (𝑋(1st𝐹)𝑦) = ((1st𝐹)‘⟨𝑋, 𝑦⟩)
6664, 65syl6req 2816 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐹)‘⟨𝑋, 𝑦⟩) = ((1st𝐾)‘𝑦))
671, 2, 61, 62, 63, 6, 51, 8, 55curf11 17132 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐾)‘𝑧) = (𝑋(1st𝐹)𝑧))
68 df-ov 6845 . . . . . . . . . . 11 (𝑋(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩)
6967, 68syl6req 2816 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐹)‘⟨𝑋, 𝑧⟩) = ((1st𝐾)‘𝑧))
7066, 69oveq12d 6860 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((1st𝐹)‘⟨𝑋, 𝑦⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑋, 𝑧⟩)) = (((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
7160, 70feq23d 6218 . . . . . . . 8 (((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 223 . . . . . . 7 (((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, 51catidcl 16608 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
74 op1stg 7378 . . . . . . . . . 10 ((𝑋𝐴𝑦𝐵) → (1st ‘⟨𝑋, 𝑦⟩) = 𝑋)
7551, 52, 74syl2anc 579 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘⟨𝑋, 𝑦⟩) = 𝑋)
76 op1stg 7378 . . . . . . . . . 10 ((𝑋𝐴𝑧𝐵) → (1st ‘⟨𝑋, 𝑧⟩) = 𝑋)
7751, 55, 76syl2anc 579 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘⟨𝑋, 𝑧⟩) = 𝑋)
7875, 77oveq12d 6860 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)) = (𝑋(Hom ‘𝐶)𝑋))
7973, 78eleqtrrd 2847 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑋) ∈ ((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)))
80 simpr 477 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
81 op2ndg 7379 . . . . . . . . . 10 ((𝑋𝐴𝑦𝐵) → (2nd ‘⟨𝑋, 𝑦⟩) = 𝑦)
8251, 52, 81syl2anc 579 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘⟨𝑋, 𝑦⟩) = 𝑦)
83 op2ndg 7379 . . . . . . . . . 10 ((𝑋𝐴𝑧𝐵) → (2nd ‘⟨𝑋, 𝑧⟩) = 𝑧)
8451, 55, 83syl2anc 579 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘⟨𝑋, 𝑧⟩) = 𝑧)
8582, 84oveq12d 6860 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)) = (𝑦(Hom ‘𝐷)𝑧))
8680, 85eleqtrrd 2847 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)))
8772, 79, 86fovrnd 7004 . . . . . 6 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) ∈ (((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
8887fmpttd 6575 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)):(𝑦(Hom ‘𝐷)𝑧)⟶(((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
8918oveqd 6859 . . . . . . 7 (𝜑 → (𝑦(2nd𝐾)𝑧) = (𝑦(𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))𝑧))
9043ovmpt4g 6981 . . . . . . . 8 ((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)))
9145, 90mp3an3 1574 . . . . . . 7 ((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)))
9289, 91sylan9eq 2819 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → (𝑦(2nd𝐾)𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))
9392feq1d 6208 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → ((𝑦(2nd𝐾)𝑧):(𝑦(Hom ‘𝐷)𝑧)⟶(((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)) ↔ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)):(𝑦(Hom ‘𝐷)𝑧)⟶(((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧))))
9488, 93mpbird 248 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → (𝑦(2nd𝐾)𝑧):(𝑦(Hom ‘𝐷)𝑧)⟶(((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
953adantr 472 . . . . . . . . 9 ((𝜑𝑦𝐵) → 𝐶 ∈ Cat)
964adantr 472 . . . . . . . . 9 ((𝜑𝑦𝐵) → 𝐷 ∈ Cat)
97 eqid 2765 . . . . . . . . 9 (Id‘(𝐶 ×c 𝐷)) = (Id‘(𝐶 ×c 𝐷))
9830, 95, 96, 2, 6, 10, 23, 97, 37, 38xpcid 17095 . . . . . . . 8 ((𝜑𝑦𝐵) → ((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩) = ⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩)
9998fveq2d 6379 . . . . . . 7 ((𝜑𝑦𝐵) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩))
100 df-ov 6845 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩)
10199, 100syl6eqr 2817 . . . . . 6 ((𝜑𝑦𝐵) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)))
10234adantr 472 . . . . . . 7 ((𝜑𝑦𝐵) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
1037, 53sylan 575 . . . . . . 7 ((𝜑𝑦𝐵) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
10431, 97, 24, 102, 103funcid 16795 . . . . . 6 ((𝜑𝑦𝐵) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑋, 𝑦⟩)))
105101, 104eqtr3d 2801 . . . . 5 ((𝜑𝑦𝐵) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑋, 𝑦⟩)))
1065adantr 472 . . . . . 6 ((𝜑𝑦𝐵) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
1076, 9, 23, 96, 38catidcl 16608 . . . . . 6 ((𝜑𝑦𝐵) → ((Id‘𝐷)‘𝑦) ∈ (𝑦(Hom ‘𝐷)𝑦))
1081, 2, 95, 96, 106, 6, 37, 8, 38, 9, 10, 38, 107curf12 17133 . . . . 5 ((𝜑𝑦𝐵) → ((𝑦(2nd𝐾)𝑦)‘((Id‘𝐷)‘𝑦)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)))
1091, 2, 95, 96, 106, 6, 37, 8, 38curf11 17132 . . . . . . 7 ((𝜑𝑦𝐵) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
110109, 65syl6eq 2815 . . . . . 6 ((𝜑𝑦𝐵) → ((1st𝐾)‘𝑦) = ((1st𝐹)‘⟨𝑋, 𝑦⟩))
111110fveq2d 6379 . . . . 5 ((𝜑𝑦𝐵) → ((Id‘𝐸)‘((1st𝐾)‘𝑦)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑋, 𝑦⟩)))
112105, 108, 1113eqtr4d 2809 . . . 4 ((𝜑𝑦𝐵) → ((𝑦(2nd𝐾)𝑦)‘((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st𝐾)‘𝑦)))
11373ad2ant1 1163 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋𝐴)
114 simp21 1263 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑦𝐵)
115 simp22 1264 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧𝐵)
116 eqid 2765 . . . . . . . . . 10 (comp‘𝐶) = (comp‘𝐶)
117 eqid 2765 . . . . . . . . . 10 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
118 simp23 1265 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤𝐵)
11933ad2ant1 1163 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat)
1202, 59, 10, 119, 113catidcl 16608 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
121 simp3l 1258 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
122 simp3r 1259 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ∈ (𝑧(Hom ‘𝐷)𝑤))
12330, 2, 6, 59, 9, 113, 114, 113, 115, 116, 25, 117, 113, 118, 120, 121, 120, 122xpcco2 17093 . . . . . . . . 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)⟩)
1242, 59, 10, 119, 113, 116, 113, 120catlid 16609 . . . . . . . . . 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))
125124opeq1d 4565 . . . . . . . . 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)⟩)
126123, 125eqtrd 2799 . . . . . . . 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)⟩)
127126fveq2d 6379 . . . . . . 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)⟩))
128 df-ov 6845 . . . . . . 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)⟩)
129127, 128syl6eqr 2817 . . . . . 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)))
130343ad2ant1 1163 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
131113, 114, 53syl2anc 579 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
132113, 115, 56syl2anc 579 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
133 opelxpi 5314 . . . . . . . 8 ((𝑋𝐴𝑤𝐵) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
134113, 118, 133syl2anc 579 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
135 opelxpi 5314 . . . . . . . . 9 ((((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨((Id‘𝐶)‘𝑋), 𝑔⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑦(Hom ‘𝐷)𝑧)))
136120, 121, 135syl2anc 579 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑔⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑦(Hom ‘𝐷)𝑧)))
13730, 2, 6, 59, 9, 113, 114, 113, 115, 49xpchom2 17092 . . . . . . . 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)))
138136, 137eleqtrrd 2847 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑔⟩ ∈ (⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩))
139 opelxpi 5314 . . . . . . . . 9 ((((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤)) → ⟨((Id‘𝐶)‘𝑋), ⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
140120, 122, 139syl2anc 579 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), ⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
14130, 2, 6, 59, 9, 113, 115, 113, 118, 49xpchom2 17092 . . . . . . . 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)))
142140, 141eleqtrrd 2847 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), ⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩))
14331, 49, 117, 26, 130, 131, 132, 134, 138, 142funcco 16796 . . . . . 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⟩)))
144129, 143eqtr3d 2801 . . . . 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⟩)))
14543ad2ant1 1163 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat)
14653ad2ant1 1163 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
1476, 9, 25, 145, 114, 115, 118, 121, 122catcocl 16611 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐷)𝑤))
1481, 2, 119, 145, 146, 6, 113, 8, 114, 9, 10, 118, 147curf12 17133 . . . . 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)))
1491, 2, 119, 145, 146, 6, 113, 8, 114curf11 17132 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
150149, 65syl6eq 2815 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑦) = ((1st𝐹)‘⟨𝑋, 𝑦⟩))
1511, 2, 119, 145, 146, 6, 113, 8, 115curf11 17132 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑧) = (𝑋(1st𝐹)𝑧))
152151, 68syl6eq 2815 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
153150, 152opeq12d 4567 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1st𝐾)‘𝑦), ((1st𝐾)‘𝑧)⟩ = ⟨((1st𝐹)‘⟨𝑋, 𝑦⟩), ((1st𝐹)‘⟨𝑋, 𝑧⟩)⟩)
1541, 2, 119, 145, 146, 6, 113, 8, 118curf11 17132 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑤) = (𝑋(1st𝐹)𝑤))
155 df-ov 6845 . . . . . . . 8 (𝑋(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩)
156154, 155syl6eq 2815 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩))
157153, 156oveq12d 6860 . . . . . 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⟩)))
1581, 2, 119, 145, 146, 6, 113, 8, 115, 9, 10, 118, 122curf12 17133 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd𝐾)𝑤)‘) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)))
159 df-ov 6845 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ⟩)
160158, 159syl6eq 2815 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd𝐾)𝑤)‘) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ⟩))
1611, 2, 119, 145, 146, 6, 113, 8, 114, 9, 10, 115, 121curf12 17133 . . . . . . 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))
162 df-ov 6845 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩)
163161, 162syl6eq 2815 . . . . . 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⟩))
164157, 160, 163oveq123d 6863 . . . . 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⟩)))
165144, 148, 1643eqtr4d 2809 . . . 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)))
1666, 21, 9, 22, 23, 24, 25, 26, 4, 29, 42, 48, 94, 112, 165isfuncd 16790 . . 3 (𝜑 → (1st𝐾)(𝐷 Func 𝐸)(2nd𝐾))
167 df-br 4810 . . 3 ((1st𝐾)(𝐷 Func 𝐸)(2nd𝐾) ↔ ⟨(1st𝐾), (2nd𝐾)⟩ ∈ (𝐷 Func 𝐸))
168166, 167sylib 209 . 2 (𝜑 → ⟨(1st𝐾), (2nd𝐾)⟩ ∈ (𝐷 Func 𝐸))
16920, 168eqeltrd 2844 1 (𝜑𝐾 ∈ (𝐷 Func 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  Vcvv 3350  cop 4340   class class class wbr 4809  cmpt 4888   × cxp 5275  Rel wrel 5282   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  cmpt2 6844  1st c1st 7364  2nd c2nd 7365  Basecbs 16130  Hom chom 16225  compcco 16226  Catccat 16590  Idccid 16591   Func cfunc 16779   ×c cxpc 17074   curryF ccurf 17116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-hom 16238  df-cco 16239  df-cat 16594  df-cid 16595  df-func 16783  df-xpc 17078  df-curf 17120
This theorem is referenced by:  curf2cl  17137  curfcl  17138
  Copyright terms: Public domain W3C validator