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Theorem lpss3 22870
Description: Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
lpss3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ ((limPtβ€˜π½)β€˜π‘‡) βŠ† ((limPtβ€˜π½)β€˜π‘†))
Proof of Theorem lpss3
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 simp1 1134 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ 𝐽 ∈ Top)
2 simp2 1135 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ 𝑆 βŠ† 𝑋)
32ssdifssd 4143 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ (𝑆 βˆ– {π‘₯}) βŠ† 𝑋)
4 simp3 1136 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ 𝑇 βŠ† 𝑆)
54ssdifd 4141 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ (𝑇 βˆ– {π‘₯}) βŠ† (𝑆 βˆ– {π‘₯}))
6 lpfval.1 . . . . . 6 𝑋 = βˆͺ 𝐽
76clsss 22780 . . . . 5 ((𝐽 ∈ Top ∧ (𝑆 βˆ– {π‘₯}) βŠ† 𝑋 ∧ (𝑇 βˆ– {π‘₯}) βŠ† (𝑆 βˆ– {π‘₯})) β†’ ((clsβ€˜π½)β€˜(𝑇 βˆ– {π‘₯})) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))
81, 3, 5, 7syl3anc 1369 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ ((clsβ€˜π½)β€˜(𝑇 βˆ– {π‘₯})) βŠ† ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})))
98sseld 3982 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑇 βˆ– {π‘₯})) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
104, 2sstrd 3993 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ 𝑇 βŠ† 𝑋)
116islp 22866 . . . 4 ((𝐽 ∈ Top ∧ 𝑇 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘‡) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑇 βˆ– {π‘₯}))))
121, 10, 11syl2anc 582 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘‡) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑇 βˆ– {π‘₯}))))
136islp 22866 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
141, 2, 13syl2anc 582 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
159, 12, 143imtr4d 293 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘‡) β†’ π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†)))
1615ssrdv 3989 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ ((limPtβ€˜π½)β€˜π‘‡) βŠ† ((limPtβ€˜π½)β€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   βˆ– cdif 3946   βŠ† wss 3949  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22617  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-cls 22747  df-lp 22862
This theorem is referenced by:  perfdvf  25654  pibt2  36603  lpss2  36927  fourierdlem113  45235
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