Theorem lpss3 21995

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 ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((limPt‘𝐽)‘𝑇) ⊆ ((limPt‘𝐽)‘𝑆))

Proof of Theorem lpss3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1138	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝐽 ∈ Top)
2		simp2 1139	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ 𝑋)
3	2	ssdifssd 4043	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
4		simp3 1140	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑆)
5	4	ssdifd 4041	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥}))
6		lpfval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
7	6	clsss 21905	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋 ∧ (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥})) → ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
8	1, 3, 5, 7	syl3anc 1373	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
9	8	sseld 3886	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
10	4, 2	sstrd 3897	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋)
11	6	islp 21991	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥}))))
12	1, 10, 11	syl2anc 587	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥}))))
13	6	islp 21991	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
14	1, 2, 13	syl2anc 587	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
15	9, 12, 14	3imtr4d 297	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) → 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
16	15	ssrdv 3893	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((limPt‘𝐽)‘𝑇) ⊆ ((limPt‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ∖ cdif 3850   ⊆ wss 3853  {csn 4527  ∪ cuni 4805  ‘cfv 6358  Topctop 21744  clsccl 21869  limPtclp 21985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-iin 4893  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-top 21745  df-cld 21870  df-cls 21872  df-lp 21987

This theorem is referenced by:  perfdvf  24754  pibt2  35274  lpss2  35598  fourierdlem113  43378