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Theorem mrccls 21406
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 2771 . . 3 𝐽 = 𝐽
21clsfval 21352 . 2 (𝐽 ∈ Top → (cls‘𝐽) = (𝑎 ∈ 𝒫 𝐽 {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎𝑏}))
31cldmre 21405 . . 3 (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
4 mrccls.f . . . 4 𝐹 = (mrCls‘(Clsd‘𝐽))
54mrcfval 16749 . . 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 2810 1 (𝐽 ∈ Top → (cls‘𝐽) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  wcel 2051  {crab 3085  wss 3822  𝒫 cpw 4416   cuni 4708   cint 4745  cmpt 5004  cfv 6185  Moorecmre 16723  mrClscmrc 16724  Topctop 21220  Clsdccld 21343  clsccl 21345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-iin 4791  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-mre 16727  df-mrc 16728  df-top 21221  df-cld 21346  df-cls 21348
This theorem is referenced by:  istopclsd  38730
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