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Theorem mrcf 17145
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 17142 . 2 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)
2 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
32mrcfval 17144 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
43feq1d 6552 . 2 (𝐶 ∈ (Moore‘𝑋) → (𝐹:𝒫 𝑋𝐶 ↔ (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶))
51, 4mpbird 260 1 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2112  {crab 3068  wss 3883  𝒫 cpw 4530   cint 4876  cmpt 5152  wf 6397  cfv 6401  Moorecmre 17118  mrClscmrc 17119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5209  ax-nul 5216  ax-pow 5275  ax-pr 5339  ax-un 7545
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5153  df-id 5472  df-xp 5575  df-rel 5576  df-cnv 5577  df-co 5578  df-dm 5579  df-rn 5580  df-res 5581  df-ima 5582  df-iota 6359  df-fun 6403  df-fn 6404  df-f 6405  df-fv 6409  df-mre 17122  df-mrc 17123
This theorem is referenced by:  mrccl  17147  mrcssv  17150  mrcuni  17157  mrcun  17158  isacs2  17189  isacs4lem  18083  isacs5  18087  ismrcd2  40272  ismrc  40274  isnacs2  40279  isnacs3  40283
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