Theorem rabelpw 5298

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 4028	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		elpw2g 5295	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴))
3	1, 2	mpbiri 261	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  {crab 3413   ⊆ wss 3899  𝒫 cpw 4557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-in 3906  df-ss 3916  df-pw 4559

This theorem is used by:  rabexg  5299  pwnss  5313  satfvsuclem2  36104