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

Theorem mrcf 16392
Description: The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrcf (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)

Proof of Theorem mrcf
Dummy variables 𝑥 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mrcflem 16389 . 2 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)
2 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
32mrcfval 16391 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
43feq1d 6143 . 2 (𝐶 ∈ (Moore‘𝑋) → (𝐹:𝒫 𝑋𝐶 ↔ (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶))
51, 4mpbird 247 1 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103  {crab 3018  wss 3680  𝒫 cpw 4266   cint 4583  cmpt 4837  wf 5997  cfv 6001  Moorecmre 16365  mrClscmrc 16366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-int 4584  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fv 6009  df-mre 16369  df-mrc 16370
This theorem is referenced by:  mrccl  16394  mrcssv  16397  mrcuni  16404  mrcun  16405  isacs2  16436  isacs4lem  17290  isacs5  17294  ismrcd2  37681  ismrc  37683  isnacs2  37688  isnacs3  37692
  Copyright terms: Public domain W3C validator