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Theorem evpmval 32810
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 3487 . . 3 (𝐷 ∈ 𝑉 β†’ 𝐷 ∈ V)
3 fveq2 6885 . . . . . 6 (𝑑 = 𝐷 β†’ (pmSgnβ€˜π‘‘) = (pmSgnβ€˜π·))
43cnveqd 5869 . . . . 5 (𝑑 = 𝐷 β†’ β—‘(pmSgnβ€˜π‘‘) = β—‘(pmSgnβ€˜π·))
54imaeq1d 6052 . . . 4 (𝑑 = 𝐷 β†’ (β—‘(pmSgnβ€˜π‘‘) β€œ {1}) = (β—‘(pmSgnβ€˜π·) β€œ {1}))
6 df-evpm 19412 . . . 4 pmEven = (𝑑 ∈ V ↦ (β—‘(pmSgnβ€˜π‘‘) β€œ {1}))
7 fvex 6898 . . . . . 6 (pmSgnβ€˜π·) ∈ V
87cnvex 7915 . . . . 5 β—‘(pmSgnβ€˜π·) ∈ V
98imaex 7904 . . . 4 (β—‘(pmSgnβ€˜π·) β€œ {1}) ∈ V
105, 6, 9fvmpt 6992 . . 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 2778 1 (𝐷 ∈ 𝑉 β†’ 𝐴 = (β—‘(pmSgnβ€˜π·) β€œ {1}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3468  {csn 4623  β—‘ccnv 5668   β€œ cima 5672  β€˜cfv 6537  1c1 11113  pmSgncpsgn 19409  pmEvencevpm 19410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fv 6545  df-evpm 19412
This theorem is referenced by:  evpmsubg  32812
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