Theorem restlp 23077

Description: The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)

Hypotheses

Ref	Expression
restcls.1	⊢ 𝑋 = ∪ 𝐽
restcls.2	⊢ 𝐾 = (𝐽 ↾t 𝑌)

Assertion

Ref	Expression
restlp	⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌))

Proof of Theorem restlp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1138	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑌)
2	1	ssdifssd 4113	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑆 ∖ {𝑥}) ⊆ 𝑌)
3		restcls.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
4		restcls.2	. . . . . . 7 ⊢ 𝐾 = (𝐽 ↾t 𝑌)
5	3, 4	restcls 23075	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌))
6	2, 5	syld3an3 1411	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌))
7	6	eleq2d 2815	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) ↔ 𝑥 ∈ (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌)))
8		elin 3933	. . . 4 ⊢ (𝑥 ∈ (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥 ∈ 𝑌))
9	7, 8	bitrdi 287	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥 ∈ 𝑌)))
10		simp1 1136	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top)
11	3	toptopon 22811	. . . . . . . 8 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
12	10, 11	sylib 218	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ (TopOn‘𝑋))
13		simp2 1137	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 ⊆ 𝑋)
14		resttopon 23055	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
15	12, 13, 14	syl2anc 584	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
16	4, 15	eqeltrid 2833	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ (TopOn‘𝑌))
17		topontop 22807	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
18	16, 17	syl 17	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ Top)
19		toponuni 22808	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
20	16, 19	syl 17	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 = ∪ 𝐾)
21	1, 20	sseqtrd 3986	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ∪ 𝐾)
22		eqid 2730	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
23	22	islp 23034	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐾) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥}))))
24	18, 21, 23	syl2anc 584	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥}))))
25		elin 3933	. . . 4 ⊢ (𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌) ↔ (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ∧ 𝑥 ∈ 𝑌))
26	1, 13	sstrd 3960	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
27	3	islp 23034	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
28	10, 26, 27	syl2anc 584	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
29	28	anbi1d 631	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((𝑥 ∈ ((limPt‘𝐽)‘𝑆) ∧ 𝑥 ∈ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥 ∈ 𝑌)))
30	25, 29	bitrid 283	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥 ∈ 𝑌)))
31	9, 24, 30	3bitr4d 311	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌)))
32	31	eqrdv 2728	1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∖ cdif 3914   ∩ cin 3916   ⊆ wss 3917  {csn 4592  ∪ cuni 4874  ‘cfv 6514  (class class class)co 7390   ↾_t crest 17390  Topctop 22787  TopOnctopon 22804  clsccl 22912  limPtclp 23028

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-en 8922  df-fin 8925  df-fi 9369  df-rest 17392  df-topgen 17413  df-top 22788  df-topon 22805  df-bases 22840  df-cld 22913  df-cls 22915  df-lp 23030

This theorem is referenced by:  restperf  23078  lptioo2cn  45650  lptioo1cn  45651  limclner  45656  fourierdlem42  46154