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Theorem cldrcl 22529
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 6928 . 2 (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ dom Clsd)
2 fncld 22525 . . 3 Clsd Fn Top
32fndmi 6653 . 2 dom Clsd = Top
41, 3eleqtrdi 2843 1 (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  dom cdm 5676  cfv 6543  Topctop 22394  Clsdccld 22519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-cld 22522
This theorem is referenced by:  cldss  22532  cldopn  22534  difopn  22537  iincld  22542  uncld  22544  cldcls  22545  clsss2  22575  opncldf3  22589  restcldi  22676  restcldr  22677  paste  22797  connsubclo  22927  txcld  23106  cldregopn  35211  clddisj  47526
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