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Theorem mrcf 17653
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 17650 . 2 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)
2 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
32mrcfval 17652 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
43feq1d 6720 . 2 (𝐶 ∈ (Moore‘𝑋) → (𝐹:𝒫 𝑋𝐶 ↔ (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶))
51, 4mpbird 257 1 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  {crab 3432  wss 3962  𝒫 cpw 4604   cint 4950  cmpt 5230  wf 6558  cfv 6562  Moorecmre 17626  mrClscmrc 17627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-mre 17630  df-mrc 17631
This theorem is referenced by:  mrccl  17655  mrcssv  17658  mrcuni  17665  mrcun  17666  isacs2  17697  isacs4lem  18601  isacs5  18605  ismrcd2  42686  ismrc  42688  isnacs2  42693  isnacs3  42697
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