Theorem isf34lem1

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	isf34lem1	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝐴 ∖ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		compss.a	⊢ 𝐹 = ( 𝑥 ∈ 𝒫 𝐴 ↦ ( 𝐴 ∖ 𝑥 ) )
2		difeq2	⊢ ( 𝑥 = 𝑎 → ( 𝐴 ∖ 𝑥 ) = ( 𝐴 ∖ 𝑎 ) )
3	2	cbvmptv	⊢ ( 𝑥 ∈ 𝒫 𝐴 ↦ ( 𝐴 ∖ 𝑥 ) ) = ( 𝑎 ∈ 𝒫 𝐴 ↦ ( 𝐴 ∖ 𝑎 ) )
4	1 3	eqtri	⊢ 𝐹 = ( 𝑎 ∈ 𝒫 𝐴 ↦ ( 𝐴 ∖ 𝑎 ) )
5		difeq2	⊢ ( 𝑎 = 𝑋 → ( 𝐴 ∖ 𝑎 ) = ( 𝐴 ∖ 𝑋 ) )
6		elpw2g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴 ) )
7	6	biimpar	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ) → 𝑋 ∈ 𝒫 𝐴 )
8		difexg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∖ 𝑋 ) ∈ V )
9	8	adantr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝐴 ∖ 𝑋 ) ∈ V )
10	4 5 7 9	fvmptd3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝐴 ∖ 𝑋 ) )