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Theorem clstop 21605
Description: The closure of a topology's underlying set is the entire set. (Contributed by NM, 5-Oct-2007.) (Proof shortened by Jim Kingdon, 11-Mar-2023.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
clstop (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)

Proof of Theorem clstop
StepHypRef Expression
1 clscld.1 . . 3 𝑋 = 𝐽
21topcld 21571 . 2 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
3 cldcls 21578 . 2 (𝑋 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑋) = 𝑋)
42, 3syl 17 1 (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105   cuni 4830  cfv 6348  Topctop 21429  Clsdccld 21552  clsccl 21554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21430  df-cld 21555  df-cls 21557
This theorem is referenced by:  hauscmplem  21942
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