MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  curf1 Structured version   Visualization version   GIF version

Theorem curf1 17251
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 17250 . . 3 (𝜑 → (1st𝐺) = (𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
11 simpr 479 . . . . . 6 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
1211oveq1d 6937 . . . . 5 ((𝜑𝑥 = 𝑋) → (𝑥(1st𝐹)𝑦) = (𝑋(1st𝐹)𝑦))
1312mpteq2dv 4980 . . . 4 ((𝜑𝑥 = 𝑋) → (𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)))
14 simp1r 1212 . . . . . . . . 9 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → 𝑥 = 𝑋)
1514opeq1d 4642 . . . . . . . 8 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
1614opeq1d 4642 . . . . . . . 8 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
1715, 16oveq12d 6940 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩) = (⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩))
1814fveq2d 6450 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → ( 1𝑥) = ( 1𝑋))
19 eqidd 2778 . . . . . . 7 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → 𝑔 = 𝑔)
2017, 18, 19oveq123d 6943 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔) = (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))
2120mpteq2dv 4980 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦𝐵𝑧𝐵) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))
2221mpt2eq3dva 6996 . . . 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 4644 . . 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 5164 . . . 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 6548 . 2 (𝜑 → ((1st𝐺)‘𝑋) = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
281, 27syl5eq 2825 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 386  w3a 1071   = wceq 1601  wcel 2106  Vcvv 3397  cop 4403  cmpt 4965  cfv 6135  (class class class)co 6922  cmpt2 6924  1st c1st 7443  2nd c2nd 7444  Basecbs 16255  Hom chom 16349  Catccat 16710  Idccid 16711   Func cfunc 16899   ×c cxpc 17194   curryF ccurf 17236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-curf 17240
This theorem is referenced by:  curf11  17252  curf12  17253  curf1cl  17254  curf2ndf  17273
  Copyright terms: Public domain W3C validator