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Theorem iscld4 21195
Description: A subset is closed iff it contains its own closure. (Contributed by NM, 31-Jan-2008.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
iscld4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))

Proof of Theorem iscld4
StepHypRef Expression
1 clscld.1 . . 3 𝑋 = 𝐽
21iscld3 21194 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) = 𝑆))
31sscls 21186 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
43biantrud 528 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ⊆ 𝑆 ↔ (((cls‘𝐽)‘𝑆) ⊆ 𝑆𝑆 ⊆ ((cls‘𝐽)‘𝑆))))
5 eqss 3812 . . 3 (((cls‘𝐽)‘𝑆) = 𝑆 ↔ (((cls‘𝐽)‘𝑆) ⊆ 𝑆𝑆 ⊆ ((cls‘𝐽)‘𝑆)))
64, 5syl6rbbr 282 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) = 𝑆 ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
72, 6bitrd 271 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wss 3768   cuni 4627  cfv 6100  Topctop 21023  Clsdccld 21146  clsccl 21148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-rep 4963  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-int 4667  df-iun 4711  df-iin 4712  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-top 21024  df-cld 21149  df-cls 21151
This theorem is referenced by:  cncls2  21403  conncompcld  21563  1stckgen  21683  metcld  23429  metsscmetcld  23438
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