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Theorem bj-ismoore 36078
Description: Characterization of Moore collections. Note that there is no sethood hypothesis on 𝐴: it is implied by either side (this is obvious for the LHS, and is the content of bj-mooreset 36075 for the RHS). (Contributed by BJ, 9-Dec-2021.)
Assertion
Ref Expression
bj-ismoore (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-ismoore
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐴Moore𝐴 ∈ V)
2 bj-mooreset 36075 . 2 (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴𝐴 ∈ V)
3 pweq 4616 . . . 4 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4 unieq 4919 . . . . . 6 (𝑦 = 𝐴 𝑦 = 𝐴)
54ineq1d 4211 . . . . 5 (𝑦 = 𝐴 → ( 𝑦 𝑥) = ( 𝐴 𝑥))
6 id 22 . . . . 5 (𝑦 = 𝐴𝑦 = 𝐴)
75, 6eleq12d 2827 . . . 4 (𝑦 = 𝐴 → (( 𝑦 𝑥) ∈ 𝑦 ↔ ( 𝐴 𝑥) ∈ 𝐴))
83, 7raleqbidv 3342 . . 3 (𝑦 = 𝐴 → (∀𝑥 ∈ 𝒫 𝑦( 𝑦 𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
9 df-bj-moore 36077 . . 3 Moore = {𝑦 ∣ ∀𝑥 ∈ 𝒫 𝑦( 𝑦 𝑥) ∈ 𝑦}
108, 9elab2g 3670 . 2 (𝐴 ∈ V → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
111, 2, 10pm5.21nii 379 1 (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wcel 2106  wral 3061  Vcvv 3474  cin 3947  𝒫 cpw 4602   cuni 4908   cint 4950  Moorecmoore 36076
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323  df-pw 4604  df-uni 4909  df-int 4951  df-bj-moore 36077
This theorem is referenced by:  bj-ismoored0  36079  bj-ismoored  36080  bj-ismooredr  36082
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