Theorem rabelpw 5342

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 4090	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		elpw2g 5339	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴))
3	1, 2	mpbiri 258	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2106  {crab 3433   ⊆ wss 3963  𝒫 cpw 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607

This theorem is referenced by:  rabexg  5343  pwnss  5358  satfvsuclem2  35345