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Theorem inpw 47715
Description: Two ways of expressing a collection of subsets as seen in df-ntr 22848, unimax 4939, and others (Contributed by Zhi Wang, 27-Sep-2024.)
Assertion
Ref Expression
inpw (𝐵𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥𝐴𝑥𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉

Proof of Theorem inpw
StepHypRef Expression
1 dfin5 3949 . 2 (𝐴 ∩ 𝒫 𝐵) = {𝑥𝐴𝑥 ∈ 𝒫 𝐵}
2 elpw2g 5335 . . 3 (𝐵𝑉 → (𝑥 ∈ 𝒫 𝐵𝑥𝐵))
32rabbidv 3432 . 2 (𝐵𝑉 → {𝑥𝐴𝑥 ∈ 𝒫 𝐵} = {𝑥𝐴𝑥𝐵})
41, 3eqtrid 2776 1 (𝐵𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥𝐴𝑥𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {crab 3424  cin 3940  wss 3941  𝒫 cpw 4595
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 2695  ax-sep 5290
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-in 3948  df-ss 3958  df-pw 4597
This theorem is referenced by:  toplatglb  47838
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