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Theorem restcld 20886
Description: A closed set of a subspace topology is a closed set of the original topology intersected with the subset. (Contributed by FL, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 15-Dec-2013.)
Hypothesis
Ref Expression
restcld.1 𝑋 = 𝐽
Assertion
Ref Expression
restcld ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴 ∈ (Clsd‘(𝐽t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑆   𝑥,𝑋

Proof of Theorem restcld
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . 5 (𝑆𝑋𝑆𝑋)
2 restcld.1 . . . . . 6 𝑋 = 𝐽
32topopn 20636 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
4 ssexg 4764 . . . . 5 ((𝑆𝑋𝑋𝐽) → 𝑆 ∈ V)
51, 3, 4syl2anr 495 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ V)
6 resttop 20874 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽t 𝑆) ∈ Top)
75, 6syldan 487 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽t 𝑆) ∈ Top)
8 eqid 2621 . . . 4 (𝐽t 𝑆) = (𝐽t 𝑆)
98iscld 20741 . . 3 ((𝐽t 𝑆) ∈ Top → (𝐴 ∈ (Clsd‘(𝐽t 𝑆)) ↔ (𝐴 (𝐽t 𝑆) ∧ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆))))
107, 9syl 17 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴 ∈ (Clsd‘(𝐽t 𝑆)) ↔ (𝐴 (𝐽t 𝑆) ∧ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆))))
112restuni 20876 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 = (𝐽t 𝑆))
1211sseq2d 3612 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴𝑆𝐴 (𝐽t 𝑆)))
1311difeq1d 3705 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐴) = ( (𝐽t 𝑆) ∖ 𝐴))
1413eleq1d 2683 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆)))
1512, 14anbi12d 746 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ (𝐴 (𝐽t 𝑆) ∧ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆))))
16 elrest 16009 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)))
175, 16syldan 487 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)))
1817anbi2d 739 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ (𝐴𝑆 ∧ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆))))
192opncld 20747 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑜𝐽) → (𝑋𝑜) ∈ (Clsd‘𝐽))
2019adantlr 750 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑜𝐽) → (𝑋𝑜) ∈ (Clsd‘𝐽))
2120adantlr 750 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) → (𝑋𝑜) ∈ (Clsd‘𝐽))
2221adantr 481 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑋𝑜) ∈ (Clsd‘𝐽))
23 incom 3783 . . . . . . . . . . . . 13 (𝑋𝑆) = (𝑆𝑋)
24 df-ss 3569 . . . . . . . . . . . . . 14 (𝑆𝑋 ↔ (𝑆𝑋) = 𝑆)
2524biimpi 206 . . . . . . . . . . . . 13 (𝑆𝑋 → (𝑆𝑋) = 𝑆)
2623, 25syl5eq 2667 . . . . . . . . . . . 12 (𝑆𝑋 → (𝑋𝑆) = 𝑆)
2726ad4antlr 768 . . . . . . . . . . 11 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑋𝑆) = 𝑆)
2827difeq1d 3705 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → ((𝑋𝑆) ∖ 𝑜) = (𝑆𝑜))
29 difeq2 3700 . . . . . . . . . . . 12 ((𝑆𝐴) = (𝑜𝑆) → (𝑆 ∖ (𝑆𝐴)) = (𝑆 ∖ (𝑜𝑆)))
30 difindi 3857 . . . . . . . . . . . . 13 (𝑆 ∖ (𝑜𝑆)) = ((𝑆𝑜) ∪ (𝑆𝑆))
31 difid 3922 . . . . . . . . . . . . . 14 (𝑆𝑆) = ∅
3231uneq2i 3742 . . . . . . . . . . . . 13 ((𝑆𝑜) ∪ (𝑆𝑆)) = ((𝑆𝑜) ∪ ∅)
33 un0 3939 . . . . . . . . . . . . 13 ((𝑆𝑜) ∪ ∅) = (𝑆𝑜)
3430, 32, 333eqtri 2647 . . . . . . . . . . . 12 (𝑆 ∖ (𝑜𝑆)) = (𝑆𝑜)
3529, 34syl6eq 2671 . . . . . . . . . . 11 ((𝑆𝐴) = (𝑜𝑆) → (𝑆 ∖ (𝑆𝐴)) = (𝑆𝑜))
3635adantl 482 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑆 ∖ (𝑆𝐴)) = (𝑆𝑜))
37 dfss4 3836 . . . . . . . . . . . 12 (𝐴𝑆 ↔ (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3837biimpi 206 . . . . . . . . . . 11 (𝐴𝑆 → (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3938ad3antlr 766 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑆 ∖ (𝑆𝐴)) = 𝐴)
4028, 36, 393eqtr2rd 2662 . . . . . . . . 9 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → 𝐴 = ((𝑋𝑆) ∖ 𝑜))
4123difeq1i 3702 . . . . . . . . . 10 ((𝑋𝑆) ∖ 𝑜) = ((𝑆𝑋) ∖ 𝑜)
42 indif2 3846 . . . . . . . . . 10 (𝑆 ∩ (𝑋𝑜)) = ((𝑆𝑋) ∖ 𝑜)
43 incom 3783 . . . . . . . . . 10 (𝑆 ∩ (𝑋𝑜)) = ((𝑋𝑜) ∩ 𝑆)
4441, 42, 433eqtr2i 2649 . . . . . . . . 9 ((𝑋𝑆) ∖ 𝑜) = ((𝑋𝑜) ∩ 𝑆)
4540, 44syl6eq 2671 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → 𝐴 = ((𝑋𝑜) ∩ 𝑆))
46 ineq1 3785 . . . . . . . . . 10 (𝑥 = (𝑋𝑜) → (𝑥𝑆) = ((𝑋𝑜) ∩ 𝑆))
4746eqeq2d 2631 . . . . . . . . 9 (𝑥 = (𝑋𝑜) → (𝐴 = (𝑥𝑆) ↔ 𝐴 = ((𝑋𝑜) ∩ 𝑆)))
4847rspcev 3295 . . . . . . . 8 (((𝑋𝑜) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((𝑋𝑜) ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆))
4922, 45, 48syl2anc 692 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆))
5049ex 450 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) → ((𝑆𝐴) = (𝑜𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
5150rexlimdva 3024 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) → (∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
5251expimpd 628 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
5318, 52sylbid 230 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
54 difindi 3857 . . . . . . . . . 10 (𝑆 ∖ (𝑥𝑆)) = ((𝑆𝑥) ∪ (𝑆𝑆))
5531uneq2i 3742 . . . . . . . . . 10 ((𝑆𝑥) ∪ (𝑆𝑆)) = ((𝑆𝑥) ∪ ∅)
56 un0 3939 . . . . . . . . . 10 ((𝑆𝑥) ∪ ∅) = (𝑆𝑥)
5754, 55, 563eqtri 2647 . . . . . . . . 9 (𝑆 ∖ (𝑥𝑆)) = (𝑆𝑥)
58 difin2 3866 . . . . . . . . . 10 (𝑆𝑋 → (𝑆𝑥) = ((𝑋𝑥) ∩ 𝑆))
5958adantl 482 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝑥) = ((𝑋𝑥) ∩ 𝑆))
6057, 59syl5eq 2667 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∖ (𝑥𝑆)) = ((𝑋𝑥) ∩ 𝑆))
6160adantr 481 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥𝑆)) = ((𝑋𝑥) ∩ 𝑆))
62 simpll 789 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
635adantr 481 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑆 ∈ V)
642cldopn 20745 . . . . . . . . 9 (𝑥 ∈ (Clsd‘𝐽) → (𝑋𝑥) ∈ 𝐽)
6564adantl 482 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋𝑥) ∈ 𝐽)
66 elrestr 16010 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 ∈ V ∧ (𝑋𝑥) ∈ 𝐽) → ((𝑋𝑥) ∩ 𝑆) ∈ (𝐽t 𝑆))
6762, 63, 65, 66syl3anc 1323 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑋𝑥) ∩ 𝑆) ∈ (𝐽t 𝑆))
6861, 67eqeltrd 2698 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆))
69 inss2 3812 . . . . . 6 (𝑥𝑆) ⊆ 𝑆
7068, 69jctil 559 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆)))
71 sseq1 3605 . . . . . 6 (𝐴 = (𝑥𝑆) → (𝐴𝑆 ↔ (𝑥𝑆) ⊆ 𝑆))
72 difeq2 3700 . . . . . . 7 (𝐴 = (𝑥𝑆) → (𝑆𝐴) = (𝑆 ∖ (𝑥𝑆)))
7372eleq1d 2683 . . . . . 6 (𝐴 = (𝑥𝑆) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆)))
7471, 73anbi12d 746 . . . . 5 (𝐴 = (𝑥𝑆) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ ((𝑥𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆))))
7570, 74syl5ibrcom 237 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐴 = (𝑥𝑆) → (𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆))))
7675rexlimdva 3024 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆) → (𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆))))
7753, 76impbid 202 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
7810, 15, 773bitr2d 296 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴 ∈ (Clsd‘(𝐽t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891   cuni 4402  cfv 5847  (class class class)co 6604  t crest 16002  Topctop 20617  Clsdccld 20730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733
This theorem is referenced by:  restcldi  20887  restcldr  20888  restcls  20895  connsubclo  21137  cldllycmp  21208
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