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Theorem fnmre 17406
Description: The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 22196 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
fnmre Moore Fn V

Proof of Theorem fnmre
Dummy variables 𝑐 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vpwex 5331 . . . 4 𝒫 𝑥 ∈ V
21pwex 5334 . . 3 𝒫 𝒫 𝑥 ∈ V
32rabex 5288 . 2 {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))} ∈ V
4 df-mre 17401 . 2 Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
53, 4fnmpti 6640 1 Moore Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wne 2942  wral 3063  {crab 3406  Vcvv 3444  c0 4281  𝒫 cpw 4559   cint 4906   Fn wfn 6487  Moorecmre 17397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-fun 6494  df-fn 6495  df-mre 17401
This theorem is referenced by:  mreunirn  17416
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