Theorem cldlp 22209

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 = ∪ 𝐽

Assertion

Ref	Expression
cldlp	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((limPt‘𝐽)‘𝑆) ⊆ 𝑆))

Proof of Theorem cldlp

Step	Hyp	Ref	Expression
1		lpfval.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	iscld3 22123	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) = 𝑆))
3	1	clslp 22207	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
4	3	eqeq1d 2740	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) = 𝑆 ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = 𝑆))
5		ssequn2 4113	. . 3 ⊢ (((limPt‘𝐽)‘𝑆) ⊆ 𝑆 ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = 𝑆)
6	4, 5	bitr4di 288	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) = 𝑆 ↔ ((limPt‘𝐽)‘𝑆) ⊆ 𝑆))
7	2, 6	bitrd 278	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((limPt‘𝐽)‘𝑆) ⊆ 𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∪ cun 3881   ⊆ wss 3883  ∪ cuni 4836  ‘cfv 6418  Topctop 21950  Clsdccld 22075  clsccl 22077  limPtclp 22193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-top 21951  df-cld 22078  df-ntr 22079  df-cls 22080  df-nei 22157  df-lp 22195

This theorem is referenced by:  pibt2  35515