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Theorem restcld 21484
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 21218 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
4 ssexg 5083 . . . . 5 ((𝑆𝑋𝑋𝐽) → 𝑆 ∈ V)
51, 3, 4syl2anr 587 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ V)
6 resttop 21472 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽t 𝑆) ∈ Top)
75, 6syldan 582 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽t 𝑆) ∈ Top)
8 eqid 2779 . . . 4 (𝐽t 𝑆) = (𝐽t 𝑆)
98iscld 21339 . . 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 21474 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 = (𝐽t 𝑆))
1211sseq2d 3890 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴𝑆𝐴 (𝐽t 𝑆)))
1311difeq1d 3989 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐴) = ( (𝐽t 𝑆) ∖ 𝐴))
1413eleq1d 2851 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆)))
1512, 14anbi12d 621 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ (𝐴 (𝐽t 𝑆) ∧ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆))))
16 elrest 16557 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)))
175, 16syldan 582 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)))
1817anbi2d 619 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ (𝐴𝑆 ∧ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆))))
192opncld 21345 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → (𝑋𝑜) ∈ (Clsd‘𝐽))
2019ad5ant14 745 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑋𝑜) ∈ (Clsd‘𝐽))
21 incom 4067 . . . . . . . . . . . 12 (𝑋𝑆) = (𝑆𝑋)
22 df-ss 3844 . . . . . . . . . . . . 13 (𝑆𝑋 ↔ (𝑆𝑋) = 𝑆)
2322biimpi 208 . . . . . . . . . . . 12 (𝑆𝑋 → (𝑆𝑋) = 𝑆)
2421, 23syl5eq 2827 . . . . . . . . . . 11 (𝑆𝑋 → (𝑋𝑆) = 𝑆)
2524ad4antlr 720 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑋𝑆) = 𝑆)
2625difeq1d 3989 . . . . . . . . 9 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → ((𝑋𝑆) ∖ 𝑜) = (𝑆𝑜))
27 difeq2 3984 . . . . . . . . . . 11 ((𝑆𝐴) = (𝑜𝑆) → (𝑆 ∖ (𝑆𝐴)) = (𝑆 ∖ (𝑜𝑆)))
28 difindi 4146 . . . . . . . . . . . 12 (𝑆 ∖ (𝑜𝑆)) = ((𝑆𝑜) ∪ (𝑆𝑆))
29 difid 4217 . . . . . . . . . . . . 13 (𝑆𝑆) = ∅
3029uneq2i 4026 . . . . . . . . . . . 12 ((𝑆𝑜) ∪ (𝑆𝑆)) = ((𝑆𝑜) ∪ ∅)
31 un0 4231 . . . . . . . . . . . 12 ((𝑆𝑜) ∪ ∅) = (𝑆𝑜)
3228, 30, 313eqtri 2807 . . . . . . . . . . 11 (𝑆 ∖ (𝑜𝑆)) = (𝑆𝑜)
3327, 32syl6eq 2831 . . . . . . . . . 10 ((𝑆𝐴) = (𝑜𝑆) → (𝑆 ∖ (𝑆𝐴)) = (𝑆𝑜))
3433adantl 474 . . . . . . . . 9 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑆 ∖ (𝑆𝐴)) = (𝑆𝑜))
35 dfss4 4123 . . . . . . . . . . 11 (𝐴𝑆 ↔ (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3635biimpi 208 . . . . . . . . . 10 (𝐴𝑆 → (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3736ad3antlr 718 . . . . . . . . 9 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3826, 34, 373eqtr2rd 2822 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → 𝐴 = ((𝑋𝑆) ∖ 𝑜))
3921difeq1i 3986 . . . . . . . . 9 ((𝑋𝑆) ∖ 𝑜) = ((𝑆𝑋) ∖ 𝑜)
40 indif2 4135 . . . . . . . . 9 (𝑆 ∩ (𝑋𝑜)) = ((𝑆𝑋) ∖ 𝑜)
41 incom 4067 . . . . . . . . 9 (𝑆 ∩ (𝑋𝑜)) = ((𝑋𝑜) ∩ 𝑆)
4239, 40, 413eqtr2i 2809 . . . . . . . 8 ((𝑋𝑆) ∖ 𝑜) = ((𝑋𝑜) ∩ 𝑆)
4338, 42syl6eq 2831 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → 𝐴 = ((𝑋𝑜) ∩ 𝑆))
44 ineq1 4069 . . . . . . . 8 (𝑥 = (𝑋𝑜) → (𝑥𝑆) = ((𝑋𝑜) ∩ 𝑆))
4544rspceeqv 3554 . . . . . . 7 (((𝑋𝑜) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((𝑋𝑜) ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆))
4620, 43, 45syl2anc 576 . . . . . 6 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆))
4746rexlimdva2 3233 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) → (∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
4847expimpd 446 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
4918, 48sylbid 232 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
50 difindi 4146 . . . . . . . . . 10 (𝑆 ∖ (𝑥𝑆)) = ((𝑆𝑥) ∪ (𝑆𝑆))
5129uneq2i 4026 . . . . . . . . . 10 ((𝑆𝑥) ∪ (𝑆𝑆)) = ((𝑆𝑥) ∪ ∅)
52 un0 4231 . . . . . . . . . 10 ((𝑆𝑥) ∪ ∅) = (𝑆𝑥)
5350, 51, 523eqtri 2807 . . . . . . . . 9 (𝑆 ∖ (𝑥𝑆)) = (𝑆𝑥)
54 difin2 4154 . . . . . . . . . 10 (𝑆𝑋 → (𝑆𝑥) = ((𝑋𝑥) ∩ 𝑆))
5554adantl 474 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝑥) = ((𝑋𝑥) ∩ 𝑆))
5653, 55syl5eq 2827 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∖ (𝑥𝑆)) = ((𝑋𝑥) ∩ 𝑆))
5756adantr 473 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥𝑆)) = ((𝑋𝑥) ∩ 𝑆))
58 simpll 754 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
595adantr 473 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑆 ∈ V)
602cldopn 21343 . . . . . . . . 9 (𝑥 ∈ (Clsd‘𝐽) → (𝑋𝑥) ∈ 𝐽)
6160adantl 474 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋𝑥) ∈ 𝐽)
62 elrestr 16558 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 ∈ V ∧ (𝑋𝑥) ∈ 𝐽) → ((𝑋𝑥) ∩ 𝑆) ∈ (𝐽t 𝑆))
6358, 59, 61, 62syl3anc 1351 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑋𝑥) ∩ 𝑆) ∈ (𝐽t 𝑆))
6457, 63eqeltrd 2867 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆))
65 inss2 4094 . . . . . 6 (𝑥𝑆) ⊆ 𝑆
6664, 65jctil 512 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆)))
67 sseq1 3883 . . . . . 6 (𝐴 = (𝑥𝑆) → (𝐴𝑆 ↔ (𝑥𝑆) ⊆ 𝑆))
68 difeq2 3984 . . . . . . 7 (𝐴 = (𝑥𝑆) → (𝑆𝐴) = (𝑆 ∖ (𝑥𝑆)))
6968eleq1d 2851 . . . . . 6 (𝐴 = (𝑥𝑆) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆)))
7067, 69anbi12d 621 . . . . 5 (𝐴 = (𝑥𝑆) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ ((𝑥𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆))))
7166, 70syl5ibrcom 239 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐴 = (𝑥𝑆) → (𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆))))
7271rexlimdva 3230 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆) → (𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆))))
7349, 72impbid 204 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
7410, 15, 733bitr2d 299 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴 ∈ (Clsd‘(𝐽t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wrex 3090  Vcvv 3416  cdif 3827  cun 3828  cin 3829  wss 3830  c0 4179   cuni 4712  cfv 6188  (class class class)co 6976  t crest 16550  Topctop 21205  Clsdccld 21328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-oadd 7909  df-er 8089  df-en 8307  df-fin 8310  df-fi 8670  df-rest 16552  df-topgen 16573  df-top 21206  df-topon 21223  df-bases 21258  df-cld 21331
This theorem is referenced by:  restcldi  21485  restcldr  21486  restcls  21493  connsubclo  21736  cldllycmp  21807
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