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Theorem fnmre 16453
Description: The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 20930 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 4998 . . . 4 𝒫 𝑥 ∈ V
21pwex 4997 . . 3 𝒫 𝒫 𝑥 ∈ V
32rabex 4964 . 2 {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))} ∈ V
4 df-mre 16448 . 2 Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
53, 4fnmpti 6183 1 Moore Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wne 2932  wral 3050  {crab 3054  Vcvv 3340  c0 4058  𝒫 cpw 4302   cint 4627   Fn wfn 6044  Moorecmre 16444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-fun 6051  df-fn 6052  df-mre 16448
This theorem is referenced by:  mreunirn  16463
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