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Theorem curf1 16805
Description: Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-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)
curf1.j 𝐽 = (Hom ‘𝐷)
curf1.1 1 = (Id‘𝐶)
Assertion
Ref Expression
curf1 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
Distinct variable groups:   𝑦,𝑔,𝑧, 1   𝑦,𝐴   𝐵,𝑔,𝑦,𝑧   𝐶,𝑔,𝑦,𝑧   𝐷,𝑔,𝑦,𝑧   𝜑,𝑔,𝑦,𝑧   𝑔,𝐸,𝑦,𝑧   𝑔,𝐽   𝑔,𝐾,𝑦,𝑧   𝑔,𝑋,𝑦,𝑧   𝑔,𝐹,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑧,𝑔)   𝐺(𝑦,𝑧,𝑔)   𝐽(𝑦,𝑧)

Proof of Theorem curf1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 curf1.k . 2 𝐾 = ((1st𝐺)‘𝑋)
2 curfval.g . . . 4 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
3 curfval.a . . . 4 𝐴 = (Base‘𝐶)
4 curfval.c . . . 4 (𝜑𝐶 ∈ Cat)
5 curfval.d . . . 4 (𝜑𝐷 ∈ Cat)
6 curfval.f . . . 4 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
7 curfval.b . . . 4 𝐵 = (Base‘𝐷)
8 curf1.j . . . 4 𝐽 = (Hom ‘𝐷)
9 curf1.1 . . . 4 1 = (Id‘𝐶)
102, 3, 4, 5, 6, 7, 8, 9curf1fval 16804 . . 3 (𝜑 → (1st𝐺) = (𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
11 simpr 477 . . . . . 6 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
1211oveq1d 6630 . . . . 5 ((𝜑𝑥 = 𝑋) → (𝑥(1st𝐹)𝑦) = (𝑋(1st𝐹)𝑦))
1312mpteq2dv 4715 . . . 4 ((𝜑𝑥 = 𝑋) → (𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)))
14 simp1r 1084 . . . . . . . . 9 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → 𝑥 = 𝑋)
1514opeq1d 4383 . . . . . . . 8 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
1614opeq1d 4383 . . . . . . . 8 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
1715, 16oveq12d 6633 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩) = (⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩))
1814fveq2d 6162 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ( 1𝑥) = ( 1𝑋))
19 eqidd 2622 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → 𝑔 = 𝑔)
2017, 18, 19oveq123d 6636 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔) = (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))
2120mpteq2dv 4715 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))
2221mpt2eq3dva 6684 . . . 4 ((𝜑𝑥 = 𝑋) → (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔))) = (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))))
2313, 22opeq12d 4385 . . 3 ((𝜑𝑥 = 𝑋) → ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
24 curf1.x . . 3 (𝜑𝑋𝐴)
25 opex 4903 . . . 4 ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ ∈ V
2625a1i 11 . . 3 (𝜑 → ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ ∈ V)
2710, 23, 24, 26fvmptd 6255 . 2 (𝜑 → ((1st𝐺)‘𝑋) = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
281, 27syl5eq 2667 1 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3190  cop 4161  cmpt 4683  cfv 5857  (class class class)co 6615  cmpt2 6617  1st c1st 7126  2nd c2nd 7127  Basecbs 15800  Hom chom 15892  Catccat 16265  Idccid 16266   Func cfunc 16454   ×c cxpc 16748   curryF ccurf 16790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-curf 16794
This theorem is referenced by:  curf11  16806  curf12  16807  curf1cl  16808  curf2ndf  16827
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