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Theorem curf1 18270
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 18269 . . 3 (𝜑 → (1st𝐺) = (𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
11 simpr 484 . . . . . 6 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
1211oveq1d 7446 . . . . 5 ((𝜑𝑥 = 𝑋) → (𝑥(1st𝐹)𝑦) = (𝑋(1st𝐹)𝑦))
1312mpteq2dv 5244 . . . 4 ((𝜑𝑥 = 𝑋) → (𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)))
14 simp1r 1199 . . . . . . . . 9 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → 𝑥 = 𝑋)
1514opeq1d 4879 . . . . . . . 8 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
1614opeq1d 4879 . . . . . . . 8 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
1715, 16oveq12d 7449 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩) = (⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩))
1814fveq2d 6910 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ( 1𝑥) = ( 1𝑋))
19 eqidd 2738 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → 𝑔 = 𝑔)
2017, 18, 19oveq123d 7452 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔) = (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))
2120mpteq2dv 5244 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))
2221mpoeq3dva 7510 . . . 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 4881 . . 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 5469 . . . 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 7023 . 2 (𝜑 → ((1st𝐺)‘𝑋) = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
281, 27eqtrid 2789 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 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cmpt 5225  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  Hom chom 17308  Catccat 17707  Idccid 17708   Func cfunc 17899   ×c cxpc 18213   curryF ccurf 18255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-curf 18259
This theorem is referenced by:  curf11  18271  curf12  18272  curf1cl  18273  curf2ndf  18292  tposcurf1  48999  postcofval  49059
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