Theorem psgneldm

Description: Property of being a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015)

	Ref	Expression
Hypotheses	psgneldm.g	⊢ 𝐺 = ( SymGrp ‘ 𝐷 )
	psgneldm.n	⊢ 𝑁 = ( pmSgn ‘ 𝐷 )
	psgneldm.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	psgneldm	⊢ ( 𝑃 ∈ dom 𝑁 ↔ ( 𝑃 ∈ 𝐵 ∧ dom ( 𝑃 ∖ I ) ∈ Fin ) )

Proof

Step	Hyp	Ref	Expression
1		psgneldm.g	⊢ 𝐺 = ( SymGrp ‘ 𝐷 )
2		psgneldm.n	⊢ 𝑁 = ( pmSgn ‘ 𝐷 )
3		psgneldm.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
4		difeq1	⊢ ( 𝑝 = 𝑃 → ( 𝑝 ∖ I ) = ( 𝑃 ∖ I ) )
5	4	dmeqd	⊢ ( 𝑝 = 𝑃 → dom ( 𝑝 ∖ I ) = dom ( 𝑃 ∖ I ) )
6	5	eleq1d	⊢ ( 𝑝 = 𝑃 → ( dom ( 𝑝 ∖ I ) ∈ Fin ↔ dom ( 𝑃 ∖ I ) ∈ Fin ) )
7		eqid	⊢ { 𝑝 ∈ 𝐵 ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = { 𝑝 ∈ 𝐵 ∣ dom ( 𝑝 ∖ I ) ∈ Fin }
8	1 3 7 2	psgnfn	⊢ 𝑁 Fn { 𝑝 ∈ 𝐵 ∣ dom ( 𝑝 ∖ I ) ∈ Fin }
9	8	fndmi	⊢ dom 𝑁 = { 𝑝 ∈ 𝐵 ∣ dom ( 𝑝 ∖ I ) ∈ Fin }
10	6 9	elrab2	⊢ ( 𝑃 ∈ dom 𝑁 ↔ ( 𝑃 ∈ 𝐵 ∧ dom ( 𝑃 ∖ I ) ∈ Fin ) )