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	⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1}))

Proof of Theorem evpmval

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		evpmval.1	. 2 ⊢ 𝐴 = (pmEven‘𝐷)
2		elex 3500	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V)
3		fveq2 6905	. . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷))
4	3	cnveqd 5885	. . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷))
5	4	imaeq1d 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
8	7	cnvex 7948	. . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V
9	8	imaex 7937	. . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V
10	5, 6, 9	fvmpt 7015	. . 3 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1}))
11	2, 10	syl 17	. 2 ⊢ (𝐷 ∈ 𝑉 → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1}))
12	1, 11	eqtrid 2788	1 ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1}))

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