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Theorem bj-ismoore 35389
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 35386 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 3459 . 2 (𝐴Moore𝐴 ∈ V)
2 bj-mooreset 35386 . 2 (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴𝐴 ∈ V)
3 pweq 4561 . . . 4 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4 unieq 4863 . . . . . 6 (𝑦 = 𝐴 𝑦 = 𝐴)
54ineq1d 4158 . . . . 5 (𝑦 = 𝐴 → ( 𝑦 𝑥) = ( 𝐴 𝑥))
6 id 22 . . . . 5 (𝑦 = 𝐴𝑦 = 𝐴)
75, 6eleq12d 2831 . . . 4 (𝑦 = 𝐴 → (( 𝑦 𝑥) ∈ 𝑦 ↔ ( 𝐴 𝑥) ∈ 𝐴))
83, 7raleqbidv 3315 . . 3 (𝑦 = 𝐴 → (∀𝑥 ∈ 𝒫 𝑦( 𝑦 𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
9 df-bj-moore 35388 . . 3 Moore = {𝑦 ∣ ∀𝑥 ∈ 𝒫 𝑦( 𝑦 𝑥) ∈ 𝑦}
108, 9elab2g 3621 . 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 1540  wcel 2105  wral 3061  Vcvv 3441  cin 3897  𝒫 cpw 4547   cuni 4852   cint 4894  Moorecmoore 35387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rab 3404  df-v 3443  df-dif 3901  df-in 3905  df-ss 3915  df-nul 4270  df-pw 4549  df-uni 4853  df-int 4895  df-bj-moore 35388
This theorem is referenced by:  bj-ismoored0  35390  bj-ismoored  35391  bj-ismooredr  35393
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