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Theorem lpcls 22868
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 22834 . . . . . . 7 (𝐽 ∈ Fre β†’ 𝐽 ∈ Top)
2 lpcls.1 . . . . . . . . . 10 𝑋 = βˆͺ 𝐽
32clsss3 22563 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
43ssdifssd 4143 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}) βŠ† 𝑋)
52clsss3 22563 . . . . . . . 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 592 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) βŠ† 𝑋)
71, 6sylan 581 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) βŠ† 𝑋)
87sseld 3982 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜(((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯})) β†’ π‘₯ ∈ 𝑋))
9 ssdifss 4136 . . . . . . . . . . 11 (𝑆 βŠ† 𝑋 β†’ (𝑆 βˆ– {π‘₯}) βŠ† 𝑋)
102clscld 22551 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑆 βˆ– {π‘₯}) βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∈ (Clsdβ€˜π½))
111, 9, 10syl2an 597 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∈ (Clsdβ€˜π½))
1211adantr 482 . . . . . . . . 9 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∈ (Clsdβ€˜π½))
132t1sncld 22830 . . . . . . . . . . . . 13 ((𝐽 ∈ Fre ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ (Clsdβ€˜π½))
1413adantlr 714 . . . . . . . . . . . 12 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ (Clsdβ€˜π½))
15 uncld 22545 . . . . . . . . . . . 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 585 . . . . . . . . . . 11 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ ({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))) ∈ (Clsdβ€˜π½))
172sscls 22560 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ (𝑆 βˆ– {π‘₯}) βŠ† 𝑋) β†’ (𝑆 βˆ– {π‘₯}) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))
181, 9, 17syl2an 597 . . . . . . . . . . . . 13 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βˆ– {π‘₯}) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))
19 ssundif 4488 . . . . . . . . . . . . 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 482 . . . . . . . . . . 11 (((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ 𝑆 βŠ† ({π‘₯} βˆͺ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
222clsss2 22576 . . . . . . . . . . 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 585 . . . . . . . . . 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 4488 . . . . . . . . . 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 22576 . . . . . . . . 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 585 . . . . . . . 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 3982 . . . . . . 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 414 . . . . . 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 482 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
331, 3sylan 581 . . . . . . 7 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
3433ssdifssd 4143 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}) βŠ† 𝑋)
352sscls 22560 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
361, 35sylan 581 . . . . . . 7 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
3736ssdifd 4141 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βˆ– {π‘₯}) βŠ† (((clsβ€˜π½)β€˜π‘†) βˆ– {π‘₯}))
382clsss 22558 . . . . . 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 1372 . . . . 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 3982 . . . 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 22644 . . . . 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 592 . . . 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 581 . . 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 22644 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
461, 45sylan 581 . . 3 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
4741, 44, 463bitr4d 311 . 2 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) ↔ π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†)))
4847eqrdv 2731 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 397   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3946   βˆͺ cun 3947   βŠ† wss 3949  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  Clsdccld 22520  clsccl 22522  limPtclp 22638  Frect1 22811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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 22396  df-cld 22523  df-cls 22525  df-lp 22640  df-t1 22818
This theorem is referenced by:  perfcls  22869
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