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Theorem evpmval 30780
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 3511 . . 3 (𝐷𝑉𝐷 ∈ V)
3 fveq2 6663 . . . . . 6 (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷))
43cnveqd 5739 . . . . 5 (𝑑 = 𝐷(pmSgn‘𝑑) = (pmSgn‘𝐷))
54imaeq1d 5921 . . . 4 (𝑑 = 𝐷 → ((pmSgn‘𝑑) “ {1}) = ((pmSgn‘𝐷) “ {1}))
6 df-evpm 18612 . . . 4 pmEven = (𝑑 ∈ V ↦ ((pmSgn‘𝑑) “ {1}))
7 fvex 6676 . . . . . 6 (pmSgn‘𝐷) ∈ V
87cnvex 7622 . . . . 5 (pmSgn‘𝐷) ∈ V
98imaex 7613 . . . 4 ((pmSgn‘𝐷) “ {1}) ∈ V
105, 6, 9fvmpt 6761 . . 3 (𝐷 ∈ V → (pmEven‘𝐷) = ((pmSgn‘𝐷) “ {1}))
112, 10syl 17 . 2 (𝐷𝑉 → (pmEven‘𝐷) = ((pmSgn‘𝐷) “ {1}))
121, 11syl5eq 2866 1 (𝐷𝑉𝐴 = ((pmSgn‘𝐷) “ {1}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  Vcvv 3493  {csn 4559  ccnv 5547  cima 5551  cfv 6348  1c1 10530  pmSgncpsgn 18609  pmEvencevpm 18610
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-evpm 18612
This theorem is referenced by:  evpmsubg  30782
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