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Theorem restcld 21178
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 20913 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
4 ssexg 4956 . . . . 5 ((𝑆𝑋𝑋𝐽) → 𝑆 ∈ V)
51, 3, 4syl2anr 496 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ V)
6 resttop 21166 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽t 𝑆) ∈ Top)
75, 6syldan 488 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽t 𝑆) ∈ Top)
8 eqid 2760 . . . 4 (𝐽t 𝑆) = (𝐽t 𝑆)
98iscld 21033 . . 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 21168 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 = (𝐽t 𝑆))
1211sseq2d 3774 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴𝑆𝐴 (𝐽t 𝑆)))
1311difeq1d 3870 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐴) = ( (𝐽t 𝑆) ∖ 𝐴))
1413eleq1d 2824 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆)))
1512, 14anbi12d 749 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ (𝐴 (𝐽t 𝑆) ∧ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆))))
16 elrest 16290 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)))
175, 16syldan 488 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)))
1817anbi2d 742 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ (𝐴𝑆 ∧ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆))))
192opncld 21039 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑜𝐽) → (𝑋𝑜) ∈ (Clsd‘𝐽))
2019adantlr 753 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑜𝐽) → (𝑋𝑜) ∈ (Clsd‘𝐽))
2120adantlr 753 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) → (𝑋𝑜) ∈ (Clsd‘𝐽))
2221adantr 472 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑋𝑜) ∈ (Clsd‘𝐽))
23 incom 3948 . . . . . . . . . . . . 13 (𝑋𝑆) = (𝑆𝑋)
24 df-ss 3729 . . . . . . . . . . . . . 14 (𝑆𝑋 ↔ (𝑆𝑋) = 𝑆)
2524biimpi 206 . . . . . . . . . . . . 13 (𝑆𝑋 → (𝑆𝑋) = 𝑆)
2623, 25syl5eq 2806 . . . . . . . . . . . 12 (𝑆𝑋 → (𝑋𝑆) = 𝑆)
2726ad4antlr 773 . . . . . . . . . . 11 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑋𝑆) = 𝑆)
2827difeq1d 3870 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → ((𝑋𝑆) ∖ 𝑜) = (𝑆𝑜))
29 difeq2 3865 . . . . . . . . . . . 12 ((𝑆𝐴) = (𝑜𝑆) → (𝑆 ∖ (𝑆𝐴)) = (𝑆 ∖ (𝑜𝑆)))
30 difindi 4024 . . . . . . . . . . . . 13 (𝑆 ∖ (𝑜𝑆)) = ((𝑆𝑜) ∪ (𝑆𝑆))
31 difid 4091 . . . . . . . . . . . . . 14 (𝑆𝑆) = ∅
3231uneq2i 3907 . . . . . . . . . . . . 13 ((𝑆𝑜) ∪ (𝑆𝑆)) = ((𝑆𝑜) ∪ ∅)
33 un0 4110 . . . . . . . . . . . . 13 ((𝑆𝑜) ∪ ∅) = (𝑆𝑜)
3430, 32, 333eqtri 2786 . . . . . . . . . . . 12 (𝑆 ∖ (𝑜𝑆)) = (𝑆𝑜)
3529, 34syl6eq 2810 . . . . . . . . . . 11 ((𝑆𝐴) = (𝑜𝑆) → (𝑆 ∖ (𝑆𝐴)) = (𝑆𝑜))
3635adantl 473 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑆 ∖ (𝑆𝐴)) = (𝑆𝑜))
37 dfss4 4001 . . . . . . . . . . . 12 (𝐴𝑆 ↔ (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3837biimpi 206 . . . . . . . . . . 11 (𝐴𝑆 → (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3938ad3antlr 769 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑆 ∖ (𝑆𝐴)) = 𝐴)
4028, 36, 393eqtr2rd 2801 . . . . . . . . 9 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → 𝐴 = ((𝑋𝑆) ∖ 𝑜))
4123difeq1i 3867 . . . . . . . . . 10 ((𝑋𝑆) ∖ 𝑜) = ((𝑆𝑋) ∖ 𝑜)
42 indif2 4013 . . . . . . . . . 10 (𝑆 ∩ (𝑋𝑜)) = ((𝑆𝑋) ∖ 𝑜)
43 incom 3948 . . . . . . . . . 10 (𝑆 ∩ (𝑋𝑜)) = ((𝑋𝑜) ∩ 𝑆)
4441, 42, 433eqtr2i 2788 . . . . . . . . 9 ((𝑋𝑆) ∖ 𝑜) = ((𝑋𝑜) ∩ 𝑆)
4540, 44syl6eq 2810 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → 𝐴 = ((𝑋𝑜) ∩ 𝑆))
46 ineq1 3950 . . . . . . . . . 10 (𝑥 = (𝑋𝑜) → (𝑥𝑆) = ((𝑋𝑜) ∩ 𝑆))
4746eqeq2d 2770 . . . . . . . . 9 (𝑥 = (𝑋𝑜) → (𝐴 = (𝑥𝑆) ↔ 𝐴 = ((𝑋𝑜) ∩ 𝑆)))
4847rspcev 3449 . . . . . . . 8 (((𝑋𝑜) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((𝑋𝑜) ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆))
4922, 45, 48syl2anc 696 . . . . . . 7 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆))
5049ex 449 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) → ((𝑆𝐴) = (𝑜𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
5150rexlimdva 3169 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) → (∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
5251expimpd 630 . . . 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 4024 . . . . . . . . . 10 (𝑆 ∖ (𝑥𝑆)) = ((𝑆𝑥) ∪ (𝑆𝑆))
5531uneq2i 3907 . . . . . . . . . 10 ((𝑆𝑥) ∪ (𝑆𝑆)) = ((𝑆𝑥) ∪ ∅)
56 un0 4110 . . . . . . . . . 10 ((𝑆𝑥) ∪ ∅) = (𝑆𝑥)
5754, 55, 563eqtri 2786 . . . . . . . . 9 (𝑆 ∖ (𝑥𝑆)) = (𝑆𝑥)
58 difin2 4033 . . . . . . . . . 10 (𝑆𝑋 → (𝑆𝑥) = ((𝑋𝑥) ∩ 𝑆))
5958adantl 473 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝑥) = ((𝑋𝑥) ∩ 𝑆))
6057, 59syl5eq 2806 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∖ (𝑥𝑆)) = ((𝑋𝑥) ∩ 𝑆))
6160adantr 472 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥𝑆)) = ((𝑋𝑥) ∩ 𝑆))
62 simpll 807 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
635adantr 472 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑆 ∈ V)
642cldopn 21037 . . . . . . . . 9 (𝑥 ∈ (Clsd‘𝐽) → (𝑋𝑥) ∈ 𝐽)
6564adantl 473 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋𝑥) ∈ 𝐽)
66 elrestr 16291 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 ∈ V ∧ (𝑋𝑥) ∈ 𝐽) → ((𝑋𝑥) ∩ 𝑆) ∈ (𝐽t 𝑆))
6762, 63, 65, 66syl3anc 1477 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑋𝑥) ∩ 𝑆) ∈ (𝐽t 𝑆))
6861, 67eqeltrd 2839 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆))
69 inss2 3977 . . . . . 6 (𝑥𝑆) ⊆ 𝑆
7068, 69jctil 561 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆)))
71 sseq1 3767 . . . . . 6 (𝐴 = (𝑥𝑆) → (𝐴𝑆 ↔ (𝑥𝑆) ⊆ 𝑆))
72 difeq2 3865 . . . . . . 7 (𝐴 = (𝑥𝑆) → (𝑆𝐴) = (𝑆 ∖ (𝑥𝑆)))
7372eleq1d 2824 . . . . . 6 (𝐴 = (𝑥𝑆) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆)))
7471, 73anbi12d 749 . . . . 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 3169 . . 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 383   = wceq 1632  wcel 2139  wrex 3051  Vcvv 3340  cdif 3712  cun 3713  cin 3714  wss 3715  c0 4058   cuni 4588  cfv 6049  (class class class)co 6813  t crest 16283  Topctop 20900  Clsdccld 21022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733  df-er 7911  df-en 8122  df-fin 8125  df-fi 8482  df-rest 16285  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-cld 21025
This theorem is referenced by:  restcldi  21179  restcldr  21180  restcls  21187  connsubclo  21429  cldllycmp  21500
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