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Theorem rabelpw 5345
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 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ 𝒫 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rabelpw
StepHypRef Expression
1 ssrab2 4072 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 elpw2g 5337 . 2 (𝐴𝑉 → ({𝑥𝐴𝜑} ∈ 𝒫 𝐴 ↔ {𝑥𝐴𝜑} ⊆ 𝐴))
31, 2mpbiri 258 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  {crab 3426  wss 3943  𝒫 cpw 4597
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 2697  ax-sep 5292
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599
This theorem is referenced by:  satfvsuclem2  34879
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