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Theorem clstop 20683
 Description: The closure of a topology's underlying set is entire set. (Contributed by NM, 5-Oct-2007.)
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 20649 . 2 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
3 ssid 3587 . . 3 𝑋𝑋
41iscld3 20678 . . 3 ((𝐽 ∈ Top ∧ 𝑋𝑋) → (𝑋 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑋) = 𝑋))
53, 4mpan2 703 . 2 (𝐽 ∈ Top → (𝑋 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑋) = 𝑋))
62, 5mpbid 221 1 (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630  clsccl 20632 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633  df-cls 20635 This theorem is referenced by:  hauscmplem  21019
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