Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  evpmval Structured version   Visualization version   GIF version

Theorem evpmval 32043
Description: Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
Hypothesis
Ref Expression
evpmval.1 𝐴 = (pmEvenβ€˜π·)
Assertion
Ref Expression
evpmval (𝐷 ∈ 𝑉 β†’ 𝐴 = (β—‘(pmSgnβ€˜π·) β€œ {1}))

Proof of Theorem evpmval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 evpmval.1 . 2 𝐴 = (pmEvenβ€˜π·)
2 elex 3462 . . 3 (𝐷 ∈ 𝑉 β†’ 𝐷 ∈ V)
3 fveq2 6843 . . . . . 6 (𝑑 = 𝐷 β†’ (pmSgnβ€˜π‘‘) = (pmSgnβ€˜π·))
43cnveqd 5832 . . . . 5 (𝑑 = 𝐷 β†’ β—‘(pmSgnβ€˜π‘‘) = β—‘(pmSgnβ€˜π·))
54imaeq1d 6013 . . . 4 (𝑑 = 𝐷 β†’ (β—‘(pmSgnβ€˜π‘‘) β€œ {1}) = (β—‘(pmSgnβ€˜π·) β€œ {1}))
6 df-evpm 19279 . . . 4 pmEven = (𝑑 ∈ V ↦ (β—‘(pmSgnβ€˜π‘‘) β€œ {1}))
7 fvex 6856 . . . . . 6 (pmSgnβ€˜π·) ∈ V
87cnvex 7863 . . . . 5 β—‘(pmSgnβ€˜π·) ∈ V
98imaex 7854 . . . 4 (β—‘(pmSgnβ€˜π·) β€œ {1}) ∈ V
105, 6, 9fvmpt 6949 . . 3 (𝐷 ∈ V β†’ (pmEvenβ€˜π·) = (β—‘(pmSgnβ€˜π·) β€œ {1}))
112, 10syl 17 . 2 (𝐷 ∈ 𝑉 β†’ (pmEvenβ€˜π·) = (β—‘(pmSgnβ€˜π·) β€œ {1}))
121, 11eqtrid 2785 1 (𝐷 ∈ 𝑉 β†’ 𝐴 = (β—‘(pmSgnβ€˜π·) β€œ {1}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3444  {csn 4587  β—‘ccnv 5633   β€œ cima 5637  β€˜cfv 6497  1c1 11057  pmSgncpsgn 19276  pmEvencevpm 19277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-evpm 19279
This theorem is referenced by:  evpmsubg  32045
  Copyright terms: Public domain W3C validator