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Theorem restcld 23180
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 22912 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
4 ssexg 5323 . . . . 5 ((𝑆𝑋𝑋𝐽) → 𝑆 ∈ V)
51, 3, 4syl2anr 597 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ V)
6 resttop 23168 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽t 𝑆) ∈ Top)
75, 6syldan 591 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽t 𝑆) ∈ Top)
8 eqid 2737 . . . 4 (𝐽t 𝑆) = (𝐽t 𝑆)
98iscld 23035 . . 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 23170 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 = (𝐽t 𝑆))
1211sseq2d 4016 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴𝑆𝐴 (𝐽t 𝑆)))
1311difeq1d 4125 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐴) = ( (𝐽t 𝑆) ∖ 𝐴))
1413eleq1d 2826 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆)))
1512, 14anbi12d 632 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ (𝐴 (𝐽t 𝑆) ∧ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆))))
16 elrest 17472 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)))
175, 16syldan 591 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)))
1817anbi2d 630 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ (𝐴𝑆 ∧ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆))))
192opncld 23041 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → (𝑋𝑜) ∈ (Clsd‘𝐽))
2019ad5ant14 758 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑋𝑜) ∈ (Clsd‘𝐽))
21 incom 4209 . . . . . . . . . . . 12 (𝑋𝑆) = (𝑆𝑋)
22 dfss2 3969 . . . . . . . . . . . . 13 (𝑆𝑋 ↔ (𝑆𝑋) = 𝑆)
2322biimpi 216 . . . . . . . . . . . 12 (𝑆𝑋 → (𝑆𝑋) = 𝑆)
2421, 23eqtrid 2789 . . . . . . . . . . 11 (𝑆𝑋 → (𝑋𝑆) = 𝑆)
2524ad4antlr 733 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑋𝑆) = 𝑆)
2625difeq1d 4125 . . . . . . . . 9 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → ((𝑋𝑆) ∖ 𝑜) = (𝑆𝑜))
27 difeq2 4120 . . . . . . . . . . 11 ((𝑆𝐴) = (𝑜𝑆) → (𝑆 ∖ (𝑆𝐴)) = (𝑆 ∖ (𝑜𝑆)))
28 difindi 4292 . . . . . . . . . . . 12 (𝑆 ∖ (𝑜𝑆)) = ((𝑆𝑜) ∪ (𝑆𝑆))
29 difid 4376 . . . . . . . . . . . . 13 (𝑆𝑆) = ∅
3029uneq2i 4165 . . . . . . . . . . . 12 ((𝑆𝑜) ∪ (𝑆𝑆)) = ((𝑆𝑜) ∪ ∅)
31 un0 4394 . . . . . . . . . . . 12 ((𝑆𝑜) ∪ ∅) = (𝑆𝑜)
3228, 30, 313eqtri 2769 . . . . . . . . . . 11 (𝑆 ∖ (𝑜𝑆)) = (𝑆𝑜)
3327, 32eqtrdi 2793 . . . . . . . . . 10 ((𝑆𝐴) = (𝑜𝑆) → (𝑆 ∖ (𝑆𝐴)) = (𝑆𝑜))
3433adantl 481 . . . . . . . . 9 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑆 ∖ (𝑆𝐴)) = (𝑆𝑜))
35 dfss4 4269 . . . . . . . . . . 11 (𝐴𝑆 ↔ (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3635biimpi 216 . . . . . . . . . 10 (𝐴𝑆 → (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3736ad3antlr 731 . . . . . . . . 9 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3826, 34, 373eqtr2rd 2784 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → 𝐴 = ((𝑋𝑆) ∖ 𝑜))
3921difeq1i 4122 . . . . . . . . 9 ((𝑋𝑆) ∖ 𝑜) = ((𝑆𝑋) ∖ 𝑜)
40 indif2 4281 . . . . . . . . 9 (𝑆 ∩ (𝑋𝑜)) = ((𝑆𝑋) ∖ 𝑜)
41 incom 4209 . . . . . . . . 9 (𝑆 ∩ (𝑋𝑜)) = ((𝑋𝑜) ∩ 𝑆)
4239, 40, 413eqtr2i 2771 . . . . . . . 8 ((𝑋𝑆) ∖ 𝑜) = ((𝑋𝑜) ∩ 𝑆)
4338, 42eqtrdi 2793 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → 𝐴 = ((𝑋𝑜) ∩ 𝑆))
44 ineq1 4213 . . . . . . . 8 (𝑥 = (𝑋𝑜) → (𝑥𝑆) = ((𝑋𝑜) ∩ 𝑆))
4544rspceeqv 3645 . . . . . . 7 (((𝑋𝑜) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((𝑋𝑜) ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆))
4620, 43, 45syl2anc 584 . . . . . 6 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆))
4746rexlimdva2 3157 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) → (∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
4847expimpd 453 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
4918, 48sylbid 240 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
50 difindi 4292 . . . . . . . . . 10 (𝑆 ∖ (𝑥𝑆)) = ((𝑆𝑥) ∪ (𝑆𝑆))
5129uneq2i 4165 . . . . . . . . . 10 ((𝑆𝑥) ∪ (𝑆𝑆)) = ((𝑆𝑥) ∪ ∅)
52 un0 4394 . . . . . . . . . 10 ((𝑆𝑥) ∪ ∅) = (𝑆𝑥)
5350, 51, 523eqtri 2769 . . . . . . . . 9 (𝑆 ∖ (𝑥𝑆)) = (𝑆𝑥)
54 difin2 4301 . . . . . . . . . 10 (𝑆𝑋 → (𝑆𝑥) = ((𝑋𝑥) ∩ 𝑆))
5554adantl 481 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝑥) = ((𝑋𝑥) ∩ 𝑆))
5653, 55eqtrid 2789 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∖ (𝑥𝑆)) = ((𝑋𝑥) ∩ 𝑆))
5756adantr 480 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥𝑆)) = ((𝑋𝑥) ∩ 𝑆))
58 simpll 767 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
595adantr 480 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑆 ∈ V)
602cldopn 23039 . . . . . . . . 9 (𝑥 ∈ (Clsd‘𝐽) → (𝑋𝑥) ∈ 𝐽)
6160adantl 481 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋𝑥) ∈ 𝐽)
62 elrestr 17473 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 ∈ V ∧ (𝑋𝑥) ∈ 𝐽) → ((𝑋𝑥) ∩ 𝑆) ∈ (𝐽t 𝑆))
6358, 59, 61, 62syl3anc 1373 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑋𝑥) ∩ 𝑆) ∈ (𝐽t 𝑆))
6457, 63eqeltrd 2841 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆))
65 inss2 4238 . . . . . 6 (𝑥𝑆) ⊆ 𝑆
6664, 65jctil 519 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆)))
67 sseq1 4009 . . . . . 6 (𝐴 = (𝑥𝑆) → (𝐴𝑆 ↔ (𝑥𝑆) ⊆ 𝑆))
68 difeq2 4120 . . . . . . 7 (𝐴 = (𝑥𝑆) → (𝑆𝐴) = (𝑆 ∖ (𝑥𝑆)))
6968eleq1d 2826 . . . . . 6 (𝐴 = (𝑥𝑆) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆)))
7067, 69anbi12d 632 . . . . 5 (𝐴 = (𝑥𝑆) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ ((𝑥𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆))))
7166, 70syl5ibrcom 247 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐴 = (𝑥𝑆) → (𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆))))
7271rexlimdva 3155 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆) → (𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆))))
7349, 72impbid 212 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
7410, 15, 733bitr2d 307 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴 ∈ (Clsd‘(𝐽t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333   cuni 4907  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  Clsdccld 23024
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027
This theorem is referenced by:  restcldi  23181  restcldr  23182  restcls  23189  connsubclo  23432  cldllycmp  23503  iscnrm3rlem2  48838
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