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Theorem fnmre 17636
Description: The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 22946 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 5383 . . . 4 𝒫 𝑥 ∈ V
21pwex 5386 . . 3 𝒫 𝒫 𝑥 ∈ V
32rabex 5345 . 2 {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))} ∈ V
4 df-mre 17631 . 2 Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
53, 4fnmpti 6712 1 Moore Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wne 2938  wral 3059  {crab 3433  Vcvv 3478  c0 4339  𝒫 cpw 4605   cint 4951   Fn wfn 6558  Moorecmre 17627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-fun 6565  df-fn 6566  df-mre 17631
This theorem is referenced by:  mreunirn  17646
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