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Theorem mrccls 21097
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 2806 . . 3 𝐽 = 𝐽
21clsfval 21043 . 2 (𝐽 ∈ Top → (cls‘𝐽) = (𝑎 ∈ 𝒫 𝐽 {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎𝑏}))
31cldmre 21096 . . 3 (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
4 mrccls.f . . . 4 𝐹 = (mrCls‘(Clsd‘𝐽))
54mrcfval 16473 . . 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 2843 1 (𝐽 ∈ Top → (cls‘𝐽) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  {crab 3100  wss 3769  𝒫 cpw 4351   cuni 4630   cint 4669  cmpt 4923  cfv 6101  Moorecmre 16447  mrClscmrc 16448  Topctop 20911  Clsdccld 21034  clsccl 21036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-mre 16451  df-mrc 16452  df-top 20912  df-cld 21037  df-cls 21039
This theorem is referenced by:  istopclsd  37765
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