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Theorem mrcf 16190
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 16187 . 2 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)
2 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
32mrcfval 16189 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
43feq1d 5987 . 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 1480  wcel 1987  {crab 2911  wss 3555  𝒫 cpw 4130   cint 4440  cmpt 4673  wf 5843  cfv 5847  Moorecmre 16163  mrClscmrc 16164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-mre 16167  df-mrc 16168
This theorem is referenced by:  mrccl  16192  mrcssv  16195  mrcuni  16202  mrcun  16203  isacs2  16235  isacs4lem  17089  isacs5  17093  ismrcd2  36739  ismrc  36741  isnacs2  36746  isnacs3  36750
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