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Theorem curf1 18282
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 18281 . . 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 5250 . . . 4 ((𝜑𝑥 = 𝑋) → (𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)))
14 simp1r 1197 . . . . . . . . 9 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → 𝑥 = 𝑋)
1514opeq1d 4884 . . . . . . . 8 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
1614opeq1d 4884 . . . . . . . 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 6911 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ( 1𝑥) = ( 1𝑋))
19 eqidd 2736 . . . . . . 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 5250 . . . . 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 4886 . . 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 5475 . . . 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 2787 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 1086   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cmpt 5231  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  Hom chom 17309  Catccat 17709  Idccid 17710   Func cfunc 17905   ×c cxpc 18224   curryF ccurf 18267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-curf 18271
This theorem is referenced by:  curf11  18283  curf12  18284  curf1cl  18285  curf2ndf  18304
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