MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnmre Structured version   Visualization version   GIF version

Theorem fnmre 16456
Description: The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 20942 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 5047 . . . 4 𝒫 𝑥 ∈ V
21pwex 5050 . . 3 𝒫 𝒫 𝑥 ∈ V
32rabex 5007 . 2 {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))} ∈ V
4 df-mre 16451 . 2 Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
53, 4fnmpti 6233 1 Moore Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 2156  wne 2978  wral 3096  {crab 3100  Vcvv 3391  c0 4116  𝒫 cpw 4351   cint 4669   Fn wfn 6096  Moorecmre 16447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-fun 6103  df-fn 6104  df-mre 16451
This theorem is referenced by:  mreunirn  16466
  Copyright terms: Public domain W3C validator