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Theorem bj-ismoore 33391
Description: Characterization of Moore collections among sets. (Contributed by BJ, 9-Dec-2021.)
Assertion
Ref Expression
bj-ismoore (𝐴𝑉 → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-ismoore
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pweq 4300 . . 3 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
2 unieq 4582 . . . . 5 (𝑦 = 𝐴 𝑦 = 𝐴)
32ineq1d 3964 . . . 4 (𝑦 = 𝐴 → ( 𝑦 𝑥) = ( 𝐴 𝑥))
4 id 22 . . . 4 (𝑦 = 𝐴𝑦 = 𝐴)
53, 4eleq12d 2844 . . 3 (𝑦 = 𝐴 → (( 𝑦 𝑥) ∈ 𝑦 ↔ ( 𝐴 𝑥) ∈ 𝐴))
61, 5raleqbidv 3301 . 2 (𝑦 = 𝐴 → (∀𝑥 ∈ 𝒫 𝑦( 𝑦 𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
7 df-bj-moore 33390 . 2 Moore = {𝑦 ∣ ∀𝑥 ∈ 𝒫 𝑦( 𝑦 𝑥) ∈ 𝑦}
86, 7elab2g 3504 1 (𝐴𝑉 → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  wral 3061  cin 3722  𝒫 cpw 4297   cuni 4574   cint 4611  Moorecmoore 33389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-pw 4299  df-uni 4575  df-bj-moore 33390
This theorem is referenced by:  bj-ismoorec  33392  bj-ismoored0  33393  bj-ismooredr  33396  bj-ismooredr2  33397
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