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Metamath Proof Explorer

Theorem rabelpw

Description: A restricted class abstraction is an element of the power set of its restricting set. (Contributed by AV, 9-Oct-2023)

		Ref	Expression
	Assertion	rabelpw	⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ 𝒫 𝐴 )

Step	Hyp	Ref	Expression
1		ssrab2	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ 𝐴
2		elpw2g	⊢ ( 𝐴 ∈ 𝑉 → ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ 𝒫 𝐴 ↔ { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ 𝐴 ) )
3	1 2	mpbiri	⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ 𝒫 𝐴 )