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Theorem mrcf 16858
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 16855 . 2 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)
2 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
32mrcfval 16857 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
43feq1d 6472 . 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 1538  wcel 2115  {crab 3130  wss 3910  𝒫 cpw 4512   cint 4849  cmpt 5119  wf 6324  cfv 6328  Moorecmre 16831  mrClscmrc 16832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-int 4850  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-mre 16835  df-mrc 16836
This theorem is referenced by:  mrccl  16860  mrcssv  16863  mrcuni  16870  mrcun  16871  isacs2  16902  isacs4lem  17756  isacs5  17760  ismrcd2  39437  ismrc  39439  isnacs2  39444  isnacs3  39448
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