Theorem abssexg

Description: Existence of a class of subsets. (Contributed by NM, 15-Jul-2006) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	abssexg	⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ) } ∈ V )

Proof

Step	Hyp	Ref	Expression
1		pwexg	⊢ ( 𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V )
2		df-pw	⊢ 𝒫 𝐴 = { 𝑥 ∣ 𝑥 ⊆ 𝐴 }
3	2	eleq1i	⊢ ( 𝒫 𝐴 ∈ V ↔ { 𝑥 ∣ 𝑥 ⊆ 𝐴 } ∈ V )
4		simpl	⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ) → 𝑥 ⊆ 𝐴 )
5	4	ss2abi	⊢ { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ) } ⊆ { 𝑥 ∣ 𝑥 ⊆ 𝐴 }
6		ssexg	⊢ ( ( { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ) } ⊆ { 𝑥 ∣ 𝑥 ⊆ 𝐴 } ∧ { 𝑥 ∣ 𝑥 ⊆ 𝐴 } ∈ V ) → { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ) } ∈ V )
7	5 6	mpan	⊢ ( { 𝑥 ∣ 𝑥 ⊆ 𝐴 } ∈ V → { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ) } ∈ V )
8	3 7	sylbi	⊢ ( 𝒫 𝐴 ∈ V → { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ) } ∈ V )
9	1 8	syl	⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ) } ∈ V )