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Theorem rabelpw 5219
 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 4007 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 elpw2g 5212 . 2 (𝐴𝑉 → ({𝑥𝐴𝜑} ∈ 𝒫 𝐴 ↔ {𝑥𝐴𝜑} ⊆ 𝐴))
31, 2mpbiri 261 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  {crab 3110   ⊆ wss 3881  𝒫 cpw 4497 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-sep 5168 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499 This theorem is referenced by:  satfvsuclem2  32735
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