MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscld4 Structured version   Visualization version   GIF version

Theorem iscld4 21067
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 21066 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) = 𝑆))
31sscls 21058 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
43biantrud 529 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ⊆ 𝑆 ↔ (((cls‘𝐽)‘𝑆) ⊆ 𝑆𝑆 ⊆ ((cls‘𝐽)‘𝑆))))
5 eqss 3755 . . 3 (((cls‘𝐽)‘𝑆) = 𝑆 ↔ (((cls‘𝐽)‘𝑆) ⊆ 𝑆𝑆 ⊆ ((cls‘𝐽)‘𝑆)))
64, 5syl6rbbr 279 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) = 𝑆 ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
72, 6bitrd 268 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1628  wcel 2135  wss 3711   cuni 4584  cfv 6045  Topctop 20896  Clsdccld 21018  clsccl 21020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-iin 4671  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-top 20897  df-cld 21021  df-cls 21023
This theorem is referenced by:  cncls2  21275  conncompcld  21435  1stckgen  21555  metcld  23300  cmetss  23309
  Copyright terms: Public domain W3C validator