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Theorem mrccls 23074
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 2726 . . 3 𝐽 = 𝐽
21clsfval 23020 . 2 (𝐽 ∈ Top → (cls‘𝐽) = (𝑎 ∈ 𝒫 𝐽 {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎𝑏}))
31cldmre 23073 . . 3 (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
4 mrccls.f . . . 4 𝐹 = (mrCls‘(Clsd‘𝐽))
54mrcfval 17621 . . 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 2769 1 (𝐽 ∈ Top → (cls‘𝐽) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {crab 3419  wss 3947  𝒫 cpw 4607   cuni 4913   cint 4954  cmpt 5236  cfv 6554  Moorecmre 17595  mrClscmrc 17596  Topctop 22886  Clsdccld 23011  clsccl 23013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-mre 17599  df-mrc 17600  df-top 22887  df-cld 23014  df-cls 23016
This theorem is referenced by:  istopclsd  42357
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