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Theorem fnmre 17532
Description: The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 22418 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 5375 . . . 4 𝒫 𝑥 ∈ V
21pwex 5378 . . 3 𝒫 𝒫 𝑥 ∈ V
32rabex 5332 . 2 {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))} ∈ V
4 df-mre 17527 . 2 Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
53, 4fnmpti 6691 1 Moore Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wne 2941  wral 3062  {crab 3433  Vcvv 3475  c0 4322  𝒫 cpw 4602   cint 4950   Fn wfn 6536  Moorecmre 17523
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 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-fun 6543  df-fn 6544  df-mre 17527
This theorem is referenced by:  mreunirn  17542
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