Theorem evpmval 32887

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 3492	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V)
3		fveq2 6902	. . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷))
4	3	cnveqd 5882	. . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷))
5	4	imaeq1d 6067	. . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1}))
6		df-evpm 19454	. . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1}))
7		fvex 6915	. . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V
8	7	cnvex 7939	. . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V
9	8	imaex 7928	. . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V
10	5, 6, 9	fvmpt 7010	. . 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 2780	1 ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1}))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3473  {csn 4632  ^◡ccnv 5681   “ cima 5685  ‘cfv 6553  1c1 11147  pmSgncpsgn 19451  pmEvencevpm 19452

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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-evpm 19454

This theorem is referenced by:  evpmsubg  32889