Theorem evpmval 33496

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 3479	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V)
3		fveq2 6888	. . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷))
4	3	cnveqd 5866	. . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷))
5	4	imaeq1d 6066	. . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1}))
6		df-evpm 19593	. . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1}))
7		fvex 6901	. . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V
8	7	cnvex 7931	. . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V
9	8	imaex 7920	. . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V
10	5, 6, 9	fvmpt 6996	. . 3 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1}))
11	2, 10	syl 18	. 2 ⊢ (𝐷 ∈ 𝑉 → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1}))
12	1, 11	eqtrid 2813	1 ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1}))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3458  {csn 4594  ^◡ccnv 5665   “ cima 5669  ‘cfv 6543  1c1 11119  pmSgncpsgn 19590  pmEvencevpm 19591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fv 6551  df-evpm 19593

This theorem is used by:  evpmsubg  33498