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Theorem lpss3 21752
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 1133 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝐽 ∈ Top)
2 simp2 1134 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝑆𝑋)
32ssdifssd 4073 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
4 simp3 1135 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝑇𝑆)
54ssdifd 4071 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥}))
6 lpfval.1 . . . . . 6 𝑋 = 𝐽
76clsss 21662 . . . . 5 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋 ∧ (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥})) → ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
81, 3, 5, 7syl3anc 1368 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
98sseld 3917 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
104, 2sstrd 3928 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝑇𝑋)
116islp 21748 . . . 4 ((𝐽 ∈ Top ∧ 𝑇𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥}))))
121, 10, 11syl2anc 587 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥}))))
136islp 21748 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
141, 2, 13syl2anc 587 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
159, 12, 143imtr4d 297 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) → 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
1615ssrdv 3924 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((limPt‘𝐽)‘𝑇) ⊆ ((limPt‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2112  cdif 3881  wss 3884  {csn 4528   cuni 4803  cfv 6328  Topctop 21501  clsccl 21626  limPtclp 21742
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-top 21502  df-cld 21627  df-cls 21629  df-lp 21744
This theorem is referenced by:  perfdvf  24509  pibt2  34829  lpss2  35185  fourierdlem113  42848
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