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Theorem curf1cl 16849
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 2620 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
10 eqid 2620 . . . 4 (Id‘𝐶) = (Id‘𝐶)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 16846 . . 3 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
12 fvex 6188 . . . . . . . 8 (Base‘𝐷) ∈ V
136, 12eqeltri 2695 . . . . . . 7 𝐵 ∈ V
1413mptex 6471 . . . . . 6 (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)) ∈ V
1513, 13mpt2ex 7232 . . . . . 6 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
1614, 15op1std 7163 . . . . 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)))
1711, 16syl 17 . . . 4 (𝜑 → (1st𝐾) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)))
1814, 15op2ndd 7164 . . . . 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))))
1911, 18syl 17 . . . 4 (𝜑 → (2nd𝐾) = (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))))
2017, 19opeq12d 4401 . . 3 (𝜑 → ⟨(1st𝐾), (2nd𝐾)⟩ = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
2111, 20eqtr4d 2657 . 2 (𝜑𝐾 = ⟨(1st𝐾), (2nd𝐾)⟩)
22 eqid 2620 . . . 4 (Base‘𝐸) = (Base‘𝐸)
23 eqid 2620 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
24 eqid 2620 . . . 4 (Id‘𝐷) = (Id‘𝐷)
25 eqid 2620 . . . 4 (Id‘𝐸) = (Id‘𝐸)
26 eqid 2620 . . . 4 (comp‘𝐷) = (comp‘𝐷)
27 eqid 2620 . . . 4 (comp‘𝐸) = (comp‘𝐸)
28 funcrcl 16504 . . . . . 6 (𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸) → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
295, 28syl 17 . . . . 5 (𝜑 → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat))
3029simprd 479 . . . 4 (𝜑𝐸 ∈ Cat)
31 eqid 2620 . . . . . . . . . 10 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
3231, 2, 6xpcbas 16799 . . . . . . . . 9 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
33 relfunc 16503 . . . . . . . . . 10 Rel ((𝐶 ×c 𝐷) Func 𝐸)
34 1st2ndbr 7202 . . . . . . . . . 10 ((Rel ((𝐶 ×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
3533, 5, 34sylancr 694 . . . . . . . . 9 (𝜑 → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
3632, 22, 35funcf1 16507 . . . . . . . 8 (𝜑 → (1st𝐹):(𝐴 × 𝐵)⟶(Base‘𝐸))
3736adantr 481 . . . . . . 7 ((𝜑𝑦𝐵) → (1st𝐹):(𝐴 × 𝐵)⟶(Base‘𝐸))
387adantr 481 . . . . . . 7 ((𝜑𝑦𝐵) → 𝑋𝐴)
39 simpr 477 . . . . . . 7 ((𝜑𝑦𝐵) → 𝑦𝐵)
4037, 38, 39fovrnd 6791 . . . . . 6 ((𝜑𝑦𝐵) → (𝑋(1st𝐹)𝑦) ∈ (Base‘𝐸))
41 eqid 2620 . . . . . 6 (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦))
4240, 41fmptd 6371 . . . . 5 (𝜑 → (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)):𝐵⟶(Base‘𝐸))
4317feq1d 6017 . . . . 5 (𝜑 → ((1st𝐾):𝐵⟶(Base‘𝐸) ↔ (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)):𝐵⟶(Base‘𝐸)))
4442, 43mpbird 247 . . . 4 (𝜑 → (1st𝐾):𝐵⟶(Base‘𝐸))
45 eqid 2620 . . . . . 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)))
46 ovex 6663 . . . . . . 7 (𝑦(Hom ‘𝐷)𝑧) ∈ V
4746mptex 6471 . . . . . 6 (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V
4845, 47fnmpt2i 7224 . . . . 5 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) Fn (𝐵 × 𝐵)
4919fneq1d 5969 . . . . 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)))
5048, 49mpbiri 248 . . . 4 (𝜑 → (2nd𝐾) Fn (𝐵 × 𝐵))
51 eqid 2620 . . . . . . . . 9 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
5235ad2antrr 761 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
537ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑋𝐴)
54 simplrl 799 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦𝐵)
55 opelxpi 5138 . . . . . . . . . 10 ((𝑋𝐴𝑦𝐵) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
5653, 54, 55syl2anc 692 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
57 simplrr 800 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧𝐵)
58 opelxpi 5138 . . . . . . . . . 10 ((𝑋𝐴𝑧𝐵) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
5953, 57, 58syl2anc 692 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
6032, 51, 23, 52, 56, 59funcf2 16509 . . . . . . . 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⟩)))
61 eqid 2620 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
6231, 32, 61, 9, 51, 56, 59xpchom 16801 . . . . . . . . 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⟩))))
633ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat)
644ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat)
655ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
661, 2, 63, 64, 65, 6, 53, 8, 54curf11 16847 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
67 df-ov 6638 . . . . . . . . . . 11 (𝑋(1st𝐹)𝑦) = ((1st𝐹)‘⟨𝑋, 𝑦⟩)
6866, 67syl6req 2671 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐹)‘⟨𝑋, 𝑦⟩) = ((1st𝐾)‘𝑦))
691, 2, 63, 64, 65, 6, 53, 8, 57curf11 16847 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐾)‘𝑧) = (𝑋(1st𝐹)𝑧))
70 df-ov 6638 . . . . . . . . . . 11 (𝑋(1st𝐹)𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩)
7169, 70syl6req 2671 . . . . . . . . . 10 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐹)‘⟨𝑋, 𝑧⟩) = ((1st𝐾)‘𝑧))
7268, 71oveq12d 6653 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((1st𝐹)‘⟨𝑋, 𝑦⟩)(Hom ‘𝐸)((1st𝐹)‘⟨𝑋, 𝑧⟩)) = (((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
7362, 72feq23d 6027 . . . . . . . 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))))
7460, 73mpbid 222 . . . . . . 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)))
752, 61, 10, 63, 53catidcl 16324 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
76 op1stg 7165 . . . . . . . . . 10 ((𝑋𝐴𝑦𝐵) → (1st ‘⟨𝑋, 𝑦⟩) = 𝑋)
7753, 54, 76syl2anc 692 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘⟨𝑋, 𝑦⟩) = 𝑋)
78 op1stg 7165 . . . . . . . . . 10 ((𝑋𝐴𝑧𝐵) → (1st ‘⟨𝑋, 𝑧⟩) = 𝑋)
7953, 57, 78syl2anc 692 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘⟨𝑋, 𝑧⟩) = 𝑋)
8077, 79oveq12d 6653 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)) = (𝑋(Hom ‘𝐶)𝑋))
8175, 80eleqtrrd 2702 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑋) ∈ ((1st ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐶)(1st ‘⟨𝑋, 𝑧⟩)))
82 simpr 477 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
83 op2ndg 7166 . . . . . . . . . 10 ((𝑋𝐴𝑦𝐵) → (2nd ‘⟨𝑋, 𝑦⟩) = 𝑦)
8453, 54, 83syl2anc 692 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘⟨𝑋, 𝑦⟩) = 𝑦)
85 op2ndg 7166 . . . . . . . . . 10 ((𝑋𝐴𝑧𝐵) → (2nd ‘⟨𝑋, 𝑧⟩) = 𝑧)
8653, 57, 85syl2anc 692 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘⟨𝑋, 𝑧⟩) = 𝑧)
8784, 86oveq12d 6653 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)) = (𝑦(Hom ‘𝐷)𝑧))
8882, 87eleqtrrd 2702 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ ((2nd ‘⟨𝑋, 𝑦⟩)(Hom ‘𝐷)(2nd ‘⟨𝑋, 𝑧⟩)))
8974, 81, 88fovrnd 6791 . . . . . 6 (((𝜑 ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) ∈ (((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
90 eqid 2620 . . . . . 6 (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))
9189, 90fmptd 6371 . . . . 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)))
9219oveqd 6652 . . . . . . 7 (𝜑 → (𝑦(2nd𝐾)𝑧) = (𝑦(𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))𝑧))
9345ovmpt4g 6768 . . . . . . . 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)))
9447, 93mp3an3 1411 . . . . . . 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)))
9592, 94sylan9eq 2674 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → (𝑦(2nd𝐾)𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))
9695feq1d 6017 . . . . 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))))
9791, 96mpbird 247 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → (𝑦(2nd𝐾)𝑧):(𝑦(Hom ‘𝐷)𝑧)⟶(((1st𝐾)‘𝑦)(Hom ‘𝐸)((1st𝐾)‘𝑧)))
983adantr 481 . . . . . . . . 9 ((𝜑𝑦𝐵) → 𝐶 ∈ Cat)
994adantr 481 . . . . . . . . 9 ((𝜑𝑦𝐵) → 𝐷 ∈ Cat)
100 eqid 2620 . . . . . . . . 9 (Id‘(𝐶 ×c 𝐷)) = (Id‘(𝐶 ×c 𝐷))
10131, 98, 99, 2, 6, 10, 24, 100, 38, 39xpcid 16810 . . . . . . . 8 ((𝜑𝑦𝐵) → ((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩) = ⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩)
102101fveq2d 6182 . . . . . . 7 ((𝜑𝑦𝐵) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩))
103 df-ov 6638 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘⟨((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)⟩)
104102, 103syl6eqr 2672 . . . . . 6 ((𝜑𝑦𝐵) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)))
10535adantr 481 . . . . . . 7 ((𝜑𝑦𝐵) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
1067, 55sylan 488 . . . . . . 7 ((𝜑𝑦𝐵) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
10732, 100, 25, 105, 106funcid 16511 . . . . . 6 ((𝜑𝑦𝐵) → ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)‘((Id‘(𝐶 ×c 𝐷))‘⟨𝑋, 𝑦⟩)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑋, 𝑦⟩)))
108104, 107eqtr3d 2656 . . . . 5 ((𝜑𝑦𝐵) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑋, 𝑦⟩)))
1095adantr 481 . . . . . 6 ((𝜑𝑦𝐵) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
1106, 9, 24, 99, 39catidcl 16324 . . . . . 6 ((𝜑𝑦𝐵) → ((Id‘𝐷)‘𝑦) ∈ (𝑦(Hom ‘𝐷)𝑦))
1111, 2, 98, 99, 109, 6, 38, 8, 39, 9, 10, 39, 110curf12 16848 . . . . 5 ((𝜑𝑦𝐵) → ((𝑦(2nd𝐾)𝑦)‘((Id‘𝐷)‘𝑦)) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑦⟩)((Id‘𝐷)‘𝑦)))
1121, 2, 98, 99, 109, 6, 38, 8, 39curf11 16847 . . . . . . 7 ((𝜑𝑦𝐵) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
113112, 67syl6eq 2670 . . . . . 6 ((𝜑𝑦𝐵) → ((1st𝐾)‘𝑦) = ((1st𝐹)‘⟨𝑋, 𝑦⟩))
114113fveq2d 6182 . . . . 5 ((𝜑𝑦𝐵) → ((Id‘𝐸)‘((1st𝐾)‘𝑦)) = ((Id‘𝐸)‘((1st𝐹)‘⟨𝑋, 𝑦⟩)))
115108, 111, 1143eqtr4d 2664 . . . 4 ((𝜑𝑦𝐵) → ((𝑦(2nd𝐾)𝑦)‘((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st𝐾)‘𝑦)))
11673ad2ant1 1080 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋𝐴)
117 simp21 1092 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑦𝐵)
118 simp22 1093 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧𝐵)
119 eqid 2620 . . . . . . . . . 10 (comp‘𝐶) = (comp‘𝐶)
120 eqid 2620 . . . . . . . . . 10 (comp‘(𝐶 ×c 𝐷)) = (comp‘(𝐶 ×c 𝐷))
121 simp23 1094 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤𝐵)
12233ad2ant1 1080 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat)
1232, 61, 10, 122, 116catidcl 16324 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
124 simp3l 1087 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
125 simp3r 1088 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ∈ (𝑧(Hom ‘𝐷)𝑤))
12631, 2, 6, 61, 9, 116, 117, 116, 118, 119, 26, 120, 116, 121, 123, 124, 123, 125xpcco2 16808 . . . . . . . . 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)⟩)
1272, 61, 10, 122, 116, 119, 116, 123catlid 16325 . . . . . . . . . 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))
128127opeq1d 4399 . . . . . . . . 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)⟩)
129126, 128eqtrd 2654 . . . . . . . 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)⟩)
130129fveq2d 6182 . . . . . . 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)⟩))
131 df-ov 6638 . . . . . . 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)⟩)
132130, 131syl6eqr 2672 . . . . . 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)))
133353ad2ant1 1080 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1st𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd𝐹))
134116, 117, 55syl2anc 692 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑦⟩ ∈ (𝐴 × 𝐵))
135116, 118, 58syl2anc 692 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑧⟩ ∈ (𝐴 × 𝐵))
136 opelxpi 5138 . . . . . . . 8 ((𝑋𝐴𝑤𝐵) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
137116, 121, 136syl2anc 692 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨𝑋, 𝑤⟩ ∈ (𝐴 × 𝐵))
138 opelxpi 5138 . . . . . . . . 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)))
139123, 124, 138syl2anc 692 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑔⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑦(Hom ‘𝐷)𝑧)))
14031, 2, 6, 61, 9, 116, 117, 116, 118, 51xpchom2 16807 . . . . . . . 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)))
141139, 140eleqtrrd 2702 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), 𝑔⟩ ∈ (⟨𝑋, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑧⟩))
142 opelxpi 5138 . . . . . . . . 9 ((((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤)) → ⟨((Id‘𝐶)‘𝑋), ⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
143123, 125, 142syl2anc 692 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), ⟩ ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑧(Hom ‘𝐷)𝑤)))
14431, 2, 6, 61, 9, 116, 118, 116, 121, 51xpchom2 16807 . . . . . . . 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)))
145143, 144eleqtrrd 2702 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((Id‘𝐶)‘𝑋), ⟩ ∈ (⟨𝑋, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑋, 𝑤⟩))
14632, 51, 120, 27, 133, 134, 135, 137, 141, 145funcco 16512 . . . . . 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⟩)))
147132, 146eqtr3d 2656 . . . . 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⟩)))
14843ad2ant1 1080 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat)
14953ad2ant1 1080 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
1506, 9, 26, 148, 117, 118, 121, 124, 125catcocl 16327 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐷)𝑤))
1511, 2, 122, 148, 149, 6, 116, 8, 117, 9, 10, 121, 150curf12 16848 . . . . 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)))
1521, 2, 122, 148, 149, 6, 116, 8, 117curf11 16847 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑦) = (𝑋(1st𝐹)𝑦))
153152, 67syl6eq 2670 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑦) = ((1st𝐹)‘⟨𝑋, 𝑦⟩))
1541, 2, 122, 148, 149, 6, 116, 8, 118curf11 16847 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑧) = (𝑋(1st𝐹)𝑧))
155154, 70syl6eq 2670 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑧) = ((1st𝐹)‘⟨𝑋, 𝑧⟩))
156153, 155opeq12d 4401 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ⟨((1st𝐾)‘𝑦), ((1st𝐾)‘𝑧)⟩ = ⟨((1st𝐹)‘⟨𝑋, 𝑦⟩), ((1st𝐹)‘⟨𝑋, 𝑧⟩)⟩)
1571, 2, 122, 148, 149, 6, 116, 8, 121curf11 16847 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑤) = (𝑋(1st𝐹)𝑤))
158 df-ov 6638 . . . . . . . 8 (𝑋(1st𝐹)𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩)
159157, 158syl6eq 2670 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st𝐾)‘𝑤) = ((1st𝐹)‘⟨𝑋, 𝑤⟩))
160156, 159oveq12d 6653 . . . . . 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⟩)))
1611, 2, 122, 148, 149, 6, 116, 8, 118, 9, 10, 121, 125curf12 16848 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd𝐾)𝑤)‘) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)))
162 df-ov 6638 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ⟩)
163161, 162syl6eq 2670 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑤𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd𝐾)𝑤)‘) = ((⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑋, 𝑤⟩)‘⟨((Id‘𝐶)‘𝑋), ⟩))
1641, 2, 122, 148, 149, 6, 116, 8, 117, 9, 10, 118, 124curf12 16848 . . . . . . 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))
165 df-ov 6638 . . . . . . 7 (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) = ((⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑋), 𝑔⟩)
166164, 165syl6eq 2670 . . . . . 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⟩))
167160, 163, 166oveq123d 6656 . . . . 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⟩)))
168147, 151, 1673eqtr4d 2664 . . . 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)))
1696, 22, 9, 23, 24, 25, 26, 27, 4, 30, 44, 50, 97, 115, 168isfuncd 16506 . . 3 (𝜑 → (1st𝐾)(𝐷 Func 𝐸)(2nd𝐾))
170 df-br 4645 . . 3 ((1st𝐾)(𝐷 Func 𝐸)(2nd𝐾) ↔ ⟨(1st𝐾), (2nd𝐾)⟩ ∈ (𝐷 Func 𝐸))
171169, 170sylib 208 . 2 (𝜑 → ⟨(1st𝐾), (2nd𝐾)⟩ ∈ (𝐷 Func 𝐸))
17221, 171eqeltrd 2699 1 (𝜑𝐾 ∈ (𝐷 Func 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  Vcvv 3195  cop 4174   class class class wbr 4644  cmpt 4720   × cxp 5102  Rel wrel 5109   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152  Basecbs 15838  Hom chom 15933  compcco 15934  Catccat 16306  Idccid 16307   Func cfunc 16495   ×c cxpc 16789   curryF ccurf 16831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-hom 15947  df-cco 15948  df-cat 16310  df-cid 16311  df-func 16499  df-xpc 16793  df-curf 16835
This theorem is referenced by:  curf2cl  16852  curfcl  16853
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