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Theorem iscld4 21589
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 21588 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) = 𝑆))
31sscls 21580 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
43biantrud 532 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ⊆ 𝑆 ↔ (((cls‘𝐽)‘𝑆) ⊆ 𝑆𝑆 ⊆ ((cls‘𝐽)‘𝑆))))
5 eqss 3986 . . 3 (((cls‘𝐽)‘𝑆) = 𝑆 ↔ (((cls‘𝐽)‘𝑆) ⊆ 𝑆𝑆 ⊆ ((cls‘𝐽)‘𝑆)))
64, 5syl6rbbr 291 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) = 𝑆 ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
72, 6bitrd 280 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wss 3940   cuni 4837  cfv 6352  Topctop 21417  Clsdccld 21540  clsccl 21542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-top 21418  df-cld 21543  df-cls 21545
This theorem is referenced by:  cncls2  21797  conncompcld  21958  1stckgen  22078  metcld  23824  metsscmetcld  23833
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