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Theorem lpss3 20758
 Description: Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = 𝐽
Assertion
Ref Expression
lpss3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((limPt‘𝐽)‘𝑇) ⊆ ((limPt‘𝐽)‘𝑆))

Proof of Theorem lpss3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp1 1054 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝐽 ∈ Top)
2 simp2 1055 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝑆𝑋)
32ssdifssd 3710 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
4 simp3 1056 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝑇𝑆)
54ssdifd 3708 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥}))
6 lpfval.1 . . . . . 6 𝑋 = 𝐽
76clsss 20668 . . . . 5 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋 ∧ (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥})) → ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
81, 3, 5, 7syl3anc 1318 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
98sseld 3567 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
104, 2sstrd 3578 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝑇𝑋)
116islp 20754 . . . 4 ((𝐽 ∈ Top ∧ 𝑇𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥}))))
121, 10, 11syl2anc 691 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥}))))
136islp 20754 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
141, 2, 13syl2anc 691 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
159, 12, 143imtr4d 282 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) → 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
1615ssrdv 3574 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((limPt‘𝐽)‘𝑇) ⊆ ((limPt‘𝐽)‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  clsccl 20632  limPtclp 20748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633  df-cls 20635  df-lp 20750 This theorem is referenced by:  perfdvf  23473  lpss2  32720  fourierdlem113  39112
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