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Theorem curf1 18216
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 18215 . . 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 7435 . . . . 5 ((𝜑𝑥 = 𝑋) → (𝑥(1st𝐹)𝑦) = (𝑋(1st𝐹)𝑦))
1312mpteq2dv 5250 . . . 4 ((𝜑𝑥 = 𝑋) → (𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)))
14 simp1r 1196 . . . . . . . . 9 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → 𝑥 = 𝑋)
1514opeq1d 4880 . . . . . . . 8 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
1614opeq1d 4880 . . . . . . . 8 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
1715, 16oveq12d 7438 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩) = (⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩))
1814fveq2d 6901 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ( 1𝑥) = ( 1𝑋))
19 eqidd 2729 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → 𝑔 = 𝑔)
2017, 18, 19oveq123d 7441 . . . . . 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 7497 . . . 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 4882 . . 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 5466 . . . 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 7012 . 2 (𝜑 → ((1st𝐺)‘𝑋) = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
281, 27eqtrid 2780 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 1085   = wceq 1534  wcel 2099  Vcvv 3471  cop 4635  cmpt 5231  cfv 6548  (class class class)co 7420  cmpo 7422  1st c1st 7991  2nd c2nd 7992  Basecbs 17179  Hom chom 17243  Catccat 17643  Idccid 17644   Func cfunc 17839   ×c cxpc 18158   curryF ccurf 18201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-curf 18205
This theorem is referenced by:  curf11  18217  curf12  18218  curf1cl  18219  curf2ndf  18238
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