Theorem evpmval 31520

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 3459	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V)
3		fveq2 6809	. . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷))
4	3	cnveqd 5802	. . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷))
5	4	imaeq1d 5983	. . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1}))
6		df-evpm 19167	. . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1}))
7		fvex 6822	. . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V
8	7	cnvex 7815	. . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V
9	8	imaex 7806	. . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V
10	5, 6, 9	fvmpt 6912	. . 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 2789	1 ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  Vcvv 3441  {csn 4569  ^◡ccnv 5604   “ cima 5608  ‘cfv 6463  1c1 10942  pmSgncpsgn 19164  pmEvencevpm 19165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fv 6471  df-evpm 19167

This theorem is referenced by:  evpmsubg  31522