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Theorem cldrcl 21032
Description: Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
cldrcl (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)

Proof of Theorem cldrcl
StepHypRef Expression
1 elfvdm 6381 . 2 (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ dom Clsd)
2 fncld 21028 . . 3 Clsd Fn Top
3 fndm 6151 . . 3 (Clsd Fn Top → dom Clsd = Top)
42, 3ax-mp 5 . 2 dom Clsd = Top
51, 4syl6eleq 2849 1 (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  dom cdm 5266   Fn wfn 6044  cfv 6049  Topctop 20900  Clsdccld 21022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-cld 21025
This theorem is referenced by:  cldss  21035  cldopn  21037  difopn  21040  iincld  21045  uncld  21047  cldcls  21048  clsss2  21078  opncldf3  21092  restcldi  21179  restcldr  21180  paste  21300  connsubclo  21429  txcld  21608  cldregopn  32632
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