Theorem restlp 23131

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 1139	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑌)
2	1	ssdifssd 4100	. . . . . 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 23129	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌))
6	2, 5	syld3an3 1412	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌))
7	6	eleq2d 2823	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) ↔ 𝑥 ∈ (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌)))
8		elin 3918	. . . 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 1137	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top)
11	3	toptopon 22865	. . . . . . . 8 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
12	10, 11	sylib 218	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ (TopOn‘𝑋))
13		simp2 1138	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 ⊆ 𝑋)
14		resttopon 23109	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
15	12, 13, 14	syl2anc 585	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
16	4, 15	eqeltrid 2841	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ (TopOn‘𝑌))
17		topontop 22861	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
18	16, 17	syl 17	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ Top)
19		toponuni 22862	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
20	16, 19	syl 17	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 = ∪ 𝐾)
21	1, 20	sseqtrd 3971	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ∪ 𝐾)
22		eqid 2737	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
23	22	islp 23088	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐾) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥}))))
24	18, 21, 23	syl2anc 585	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥}))))
25		elin 3918	. . . 4 ⊢ (𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌) ↔ (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ∧ 𝑥 ∈ 𝑌))
26	1, 13	sstrd 3945	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
27	3	islp 23088	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
28	10, 26, 27	syl2anc 585	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
29	28	anbi1d 632	. . . 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 2735	1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  {csn 4581  ∪ cuni 4864  ‘cfv 6493  (class class class)co 7360   ↾_t crest 17344  Topctop 22841  TopOnctopon 22858  clsccl 22966  limPtclp 23082

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-en 8888  df-fin 8891  df-fi 9318  df-rest 17346  df-topgen 17367  df-top 22842  df-topon 22859  df-bases 22894  df-cld 22967  df-cls 22969  df-lp 23084

This theorem is referenced by:  restperf  23132  lptioo2cn  45925  lptioo1cn  45926  limclner  45931  fourierdlem42  46429