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Theorem lpcls 22867
Description: The limit points of the closure of a subset are the same as the limit points of the set in a T1 space. (Contributed by Mario Carneiro, 26-Dec-2016.)
Hypothesis
Ref Expression
lpcls.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
lpcls ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) = ((limPtβ€˜π½)β€˜π‘†))
Proof of Theorem lpcls
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 t1top 22833 . . . . . . 7 (𝐽 ∈ Fre β†’ 𝐽 ∈ Top)
2 lpcls.1 . . . . . . . . . 10 𝑋 = βˆͺ 𝐽
32clsss3 22562 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
43ssdifssd 4142 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}) βŠ† 𝑋)
52clsss3 22562 . . . . . . . 8 ((𝐽 ∈ Top ∧ (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}) βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) βŠ† 𝑋)
64, 5syldan 591 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) βŠ† 𝑋)
71, 6sylan 580 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) βŠ† 𝑋)
87sseld 3981 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) β†’ π‘₯ ∈ 𝑋))
9 ssdifss 4135 . . . . . . . . . . 11 (𝑆 βŠ† 𝑋 β†’ (𝑆 βˆ– {π‘₯}) βŠ† 𝑋)
102clscld 22550 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑆 βˆ– {π‘₯}) βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∈ (Clsdβ€˜π½))
111, 9, 10syl2an 596 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∈ (Clsdβ€˜π½))
1211adantr 481 . . . . . . . . 9 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∈ (Clsdβ€˜π½))
132t1sncld 22829 . . . . . . . . . . . . 13 ((𝐽 ∈ Fre ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ (Clsdβ€˜π½))
1413adantlr 713 . . . . . . . . . . . 12 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ (Clsdβ€˜π½))
15 uncld 22544 . . . . . . . . . . . 12 (({π‘₯} ∈ (Clsdβ€˜π½) ∧ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∈ (Clsdβ€˜π½)) β†’ ({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))) ∈ (Clsdβ€˜π½))
1614, 12, 15syl2anc 584 . . . . . . . . . . 11 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ ({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))) ∈ (Clsdβ€˜π½))
172sscls 22559 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ (𝑆 βˆ– {π‘₯}) βŠ† 𝑋) β†’ (𝑆 βˆ– {π‘₯}) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))
181, 9, 17syl2an 596 . . . . . . . . . . . . 13 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βˆ– {π‘₯}) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))
19 ssundif 4487 . . . . . . . . . . . . 13 (𝑆 βŠ† ({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))) ↔ (𝑆 βˆ– {π‘₯}) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))
2018, 19sylibr 233 . . . . . . . . . . . 12 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† ({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
2120adantr 481 . . . . . . . . . . 11 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ 𝑆 βŠ† ({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
222clsss2 22575 . . . . . . . . . . 11 ((({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))) ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† ({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† ({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
2316, 21, 22syl2anc 584 . . . . . . . . . 10 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† ({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
24 ssundif 4487 . . . . . . . . . 10 (((clsβ€˜π½)β€˜π‘†) βŠ† ({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))) ↔ (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))
2523, 24sylib 217 . . . . . . . . 9 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))
262clsss2 22575 . . . . . . . . 9 ((((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∈ (Clsdβ€˜π½) ∧ (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))) β†’ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))
2712, 25, 26syl2anc 584 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))
2827sseld 3981 . . . . . . 7 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
2928ex 413 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ 𝑋 β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))))
3029com23 86 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) β†’ (π‘₯ ∈ 𝑋 β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))))
318, 30mpdd 43 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
321adantr 481 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
331, 3sylan 580 . . . . . . 7 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
3433ssdifssd 4142 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}) βŠ† 𝑋)
352sscls 22559 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
361, 35sylan 580 . . . . . . 7 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
3736ssdifd 4140 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βˆ– {π‘₯}) βŠ† (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}))
382clsss 22557 . . . . . 6 ((𝐽 ∈ Top ∧ (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}) βŠ† 𝑋 ∧ (𝑆 βˆ– {π‘₯}) βŠ† (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) β†’ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) βŠ† ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})))
3932, 34, 37, 38syl3anc 1371 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) βŠ† ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})))
4039sseld 3981 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}))))
4131, 40impbid 211 . . 3 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
422islp 22643 . . . . 5 ((𝐽 ∈ Top ∧ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}))))
433, 42syldan 591 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}))))
441, 43sylan 580 . . 3 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}))))
452islp 22643 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
461, 45sylan 580 . . 3 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
4741, 44, 463bitr4d 310 . 2 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) ↔ π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†)))
4847eqrdv 2730 1 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) = ((limPtβ€˜π½)β€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βˆ– cdif 3945   βˆͺ cun 3946   βŠ† wss 3948  {csn 4628  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22394  Clsdccld 22519  clsccl 22521  limPtclp 22637  Frect1 22810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22395  df-cld 22522  df-cls 22524  df-lp 22639  df-t1 22817
This theorem is referenced by:  perfcls  22868
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