Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ismoore Structured version   Visualization version   GIF version

Theorem bj-ismoore 37106
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 37103 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 3501 . 2 (𝐴Moore𝐴 ∈ V)
2 bj-mooreset 37103 . 2 (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴𝐴 ∈ V)
3 pweq 4614 . . . 4 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4 unieq 4918 . . . . . 6 (𝑦 = 𝐴 𝑦 = 𝐴)
54ineq1d 4219 . . . . 5 (𝑦 = 𝐴 → ( 𝑦 𝑥) = ( 𝐴 𝑥))
6 id 22 . . . . 5 (𝑦 = 𝐴𝑦 = 𝐴)
75, 6eleq12d 2835 . . . 4 (𝑦 = 𝐴 → (( 𝑦 𝑥) ∈ 𝑦 ↔ ( 𝐴 𝑥) ∈ 𝐴))
83, 7raleqbidv 3346 . . 3 (𝑦 = 𝐴 → (∀𝑥 ∈ 𝒫 𝑦( 𝑦 𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
9 df-bj-moore 37105 . . 3 Moore = {𝑦 ∣ ∀𝑥 ∈ 𝒫 𝑦( 𝑦 𝑥) ∈ 𝑦}
108, 9elab2g 3680 . 2 (𝐴 ∈ V → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
111, 2, 10pm5.21nii 378 1 (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cin 3950  𝒫 cpw 4600   cuni 4907   cint 4946  Moorecmoore 37104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-uni 4908  df-int 4947  df-bj-moore 37105
This theorem is referenced by:  bj-ismoored0  37107  bj-ismoored  37108  bj-ismooredr  37110
  Copyright terms: Public domain W3C validator