Theorem isf34lem2

Description: Lemma for isfin3-4 . (Contributed by Stefan O'Rear, 7-Nov-2014)

	Ref	Expression
Hypothesis	compss.a	⊢ 𝐹 = ( 𝑥 ∈ 𝒫 𝐴 ↦ ( 𝐴 ∖ 𝑥 ) )
Assertion	isf34lem2	⊢ ( 𝐴 ∈ 𝑉 → 𝐹 : 𝒫 𝐴 ⟶ 𝒫 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		compss.a	⊢ 𝐹 = ( 𝑥 ∈ 𝒫 𝐴 ↦ ( 𝐴 ∖ 𝑥 ) )
2		difss	⊢ ( 𝐴 ∖ 𝑥 ) ⊆ 𝐴
3		elpw2g	⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝐴 ∖ 𝑥 ) ∈ 𝒫 𝐴 ↔ ( 𝐴 ∖ 𝑥 ) ⊆ 𝐴 ) )
4	2 3	mpbiri	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∖ 𝑥 ) ∈ 𝒫 𝐴 )
5	4	adantr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝐴 ) → ( 𝐴 ∖ 𝑥 ) ∈ 𝒫 𝐴 )
6	5 1	fmptd	⊢ ( 𝐴 ∈ 𝑉 → 𝐹 : 𝒫 𝐴 ⟶ 𝒫 𝐴 )