Theorem evpmval 31412

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 3450	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V)
3		fveq2 6774	. . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷))
4	3	cnveqd 5784	. . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷))
5	4	imaeq1d 5968	. . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1}))
6		df-evpm 19100	. . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1}))
7		fvex 6787	. . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V
8	7	cnvex 7772	. . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V
9	8	imaex 7763	. . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V
10	5, 6, 9	fvmpt 6875	. . 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 2790	1 ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1}))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561  ^◡ccnv 5588   “ cima 5592  ‘cfv 6433  1c1 10872  pmSgncpsgn 19097  pmEvencevpm 19098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-evpm 19100

This theorem is referenced by:  evpmsubg  31414