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Theorem clsidm 23091
Description: The closure operation is idempotent. (Contributed by NM, 2-Oct-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
clsidm ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((cls‘𝐽)‘𝑆))

Proof of Theorem clsidm
StepHypRef Expression
1 clscld.1 . . 3 𝑋 = 𝐽
21clscld 23071 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
31clsss3 23083 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
41iscld3 23088 . . 3 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑆) ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((cls‘𝐽)‘𝑆)))
53, 4syldan 591 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((cls‘𝐽)‘𝑆)))
62, 5mpbid 232 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wss 3963   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040  clsccl 23042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-cls 23045
This theorem is referenced by:  kur14lem5  35195  opnregcld  36313
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