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Theorem cldrcl 22393
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 6880 . 2 (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ dom Clsd)
2 fncld 22389 . . 3 Clsd Fn Top
32fndmi 6607 . 2 dom Clsd = Top
41, 3eleqtrdi 2844 1 (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  dom cdm 5634  cfv 6497  Topctop 22258  Clsdccld 22383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505  df-cld 22386
This theorem is referenced by:  cldss  22396  cldopn  22398  difopn  22401  iincld  22406  uncld  22408  cldcls  22409  clsss2  22439  opncldf3  22453  restcldi  22540  restcldr  22541  paste  22661  connsubclo  22791  txcld  22970  cldregopn  34849  clddisj  47022
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