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Theorem pwelg 37684
Description: The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.)
Assertion
Ref Expression
pwelg (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem pwelg
StepHypRef Expression
1 simpr 477 . . . 4 (( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → 𝒫 𝑥𝐵)
21ralimi 2949 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ∀𝑥𝐵 𝒫 𝑥𝐵)
3 pweq 4152 . . . . 5 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
43eleq1d 2684 . . . 4 (𝑥 = 𝐴 → (𝒫 𝑥𝐵 ↔ 𝒫 𝐴𝐵))
54rspccv 3301 . . 3 (∀𝑥𝐵 𝒫 𝑥𝐵 → (𝐴𝐵 → 𝒫 𝐴𝐵))
62, 5syl 17 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 → 𝒫 𝐴𝐵))
7 simpl 473 . . . 4 (( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → 𝑥𝐵)
87ralimi 2949 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ∀𝑥𝐵 𝑥𝐵)
9 unieq 4435 . . . . . 6 (𝑥 = 𝒫 𝐴 𝑥 = 𝒫 𝐴)
10 unipw 4909 . . . . . 6 𝒫 𝐴 = 𝐴
119, 10syl6eq 2670 . . . . 5 (𝑥 = 𝒫 𝐴 𝑥 = 𝐴)
1211eleq1d 2684 . . . 4 (𝑥 = 𝒫 𝐴 → ( 𝑥𝐵𝐴𝐵))
1312rspccv 3301 . . 3 (∀𝑥𝐵 𝑥𝐵 → (𝒫 𝐴𝐵𝐴𝐵))
148, 13syl 17 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝒫 𝐴𝐵𝐴𝐵))
156, 14impbid 202 1 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  𝒫 cpw 4149   cuni 4427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-pw 4151  df-sn 4169  df-pr 4171  df-uni 4428
This theorem is referenced by:  pwinfig  37685
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