Theorem rabelpw 5356

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

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rabelpw

Step	Hyp	Ref	Expression
1		ssrab2 4075	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		elpw2g 5348	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴))
3	1, 2	mpbiri 257	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2098  {crab 3428   ⊆ wss 3947  𝒫 cpw 4604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3429  df-v 3473  df-in 3954  df-ss 3964  df-pw 4606

This theorem is referenced by:  satfvsuclem2  34975