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Theorem evpmval 33166
Description: Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
Hypothesis
Ref Expression
evpmval.1 𝐴 = (pmEven‘𝐷)
Assertion
Ref Expression
evpmval (𝐷𝑉𝐴 = ((pmSgn‘𝐷) “ {1}))

Proof of Theorem evpmval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 evpmval.1 . 2 𝐴 = (pmEven‘𝐷)
2 elex 3500 . . 3 (𝐷𝑉𝐷 ∈ V)
3 fveq2 6905 . . . . . 6 (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷))
43cnveqd 5885 . . . . 5 (𝑑 = 𝐷(pmSgn‘𝑑) = (pmSgn‘𝐷))
54imaeq1d 6076 . . . 4 (𝑑 = 𝐷 → ((pmSgn‘𝑑) “ {1}) = ((pmSgn‘𝐷) “ {1}))
6 df-evpm 19511 . . . 4 pmEven = (𝑑 ∈ V ↦ ((pmSgn‘𝑑) “ {1}))
7 fvex 6918 . . . . . 6 (pmSgn‘𝐷) ∈ V
87cnvex 7948 . . . . 5 (pmSgn‘𝐷) ∈ V
98imaex 7937 . . . 4 ((pmSgn‘𝐷) “ {1}) ∈ V
105, 6, 9fvmpt 7015 . . 3 (𝐷 ∈ V → (pmEven‘𝐷) = ((pmSgn‘𝐷) “ {1}))
112, 10syl 17 . 2 (𝐷𝑉 → (pmEven‘𝐷) = ((pmSgn‘𝐷) “ {1}))
121, 11eqtrid 2788 1 (𝐷𝑉𝐴 = ((pmSgn‘𝐷) “ {1}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  Vcvv 3479  {csn 4625  ccnv 5683  cima 5687  cfv 6560  1c1 11157  pmSgncpsgn 19508  pmEvencevpm 19509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-evpm 19511
This theorem is referenced by:  evpmsubg  33168
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