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Theorem cldlp 22876
Description: A subset of a topological space is closed iff it contains all its limit points. Corollary 6.7 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
cldlp ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ ((limPtβ€˜π½)β€˜π‘†) βŠ† 𝑆))
Proof of Theorem cldlp
StepHypRef Expression
1 lpfval.1 . . 3 𝑋 = βˆͺ 𝐽
21iscld3 22790 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ ((clsβ€˜π½)β€˜π‘†) = 𝑆))
31clslp 22874 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
43eqeq1d 2732 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π‘†) = 𝑆 ↔ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) = 𝑆))
5 ssequn2 4184 . . 3 (((limPtβ€˜π½)β€˜π‘†) βŠ† 𝑆 ↔ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) = 𝑆)
64, 5bitr4di 288 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π‘†) = 𝑆 ↔ ((limPtβ€˜π½)β€˜π‘†) βŠ† 𝑆))
72, 6bitrd 278 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 ∈ (Clsdβ€˜π½) ↔ ((limPtβ€˜π½)β€˜π‘†) βŠ† 𝑆))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   βˆͺ cun 3947   βŠ† wss 3949  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22617  Clsdccld 22742  clsccl 22744  limPtclp 22860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22618  df-cld 22745  df-ntr 22746  df-cls 22747  df-nei 22824  df-lp 22862
This theorem is referenced by:  pibt2  36603
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