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Theorem mrccls 20931
Description: Moore closure generalizes closure in a topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrccls.f 𝐹 = (mrCls‘(Clsd‘𝐽))
Assertion
Ref Expression
mrccls (𝐽 ∈ Top → (cls‘𝐽) = 𝐹)

Proof of Theorem mrccls
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . 3 𝐽 = 𝐽
21clsfval 20877 . 2 (𝐽 ∈ Top → (cls‘𝐽) = (𝑎 ∈ 𝒫 𝐽 {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎𝑏}))
31cldmre 20930 . . 3 (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
4 mrccls.f . . . 4 𝐹 = (mrCls‘(Clsd‘𝐽))
54mrcfval 16315 . . 3 ((Clsd‘𝐽) ∈ (Moore‘ 𝐽) → 𝐹 = (𝑎 ∈ 𝒫 𝐽 {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎𝑏}))
63, 5syl 17 . 2 (𝐽 ∈ Top → 𝐹 = (𝑎 ∈ 𝒫 𝐽 {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎𝑏}))
72, 6eqtr4d 2688 1 (𝐽 ∈ Top → (cls‘𝐽) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  {crab 2945  wss 3607  𝒫 cpw 4191   cuni 4468   cint 4507  cmpt 4762  cfv 5926  Moorecmre 16289  mrClscmrc 16290  Topctop 20746  Clsdccld 20868  clsccl 20870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-mre 16293  df-mrc 16294  df-top 20747  df-cld 20871  df-cls 20873
This theorem is referenced by:  istopclsd  37580
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