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Theorem restcld 20786
 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 20536 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
4 ssexg 4732 . . . . 5 ((𝑆𝑋𝑋𝐽) → 𝑆 ∈ V)
51, 3, 4syl2anr 494 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ V)
6 resttop 20774 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽t 𝑆) ∈ Top)
75, 6syldan 486 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽t 𝑆) ∈ Top)
8 eqid 2610 . . . 4 (𝐽t 𝑆) = (𝐽t 𝑆)
98iscld 20641 . . 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 20776 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 = (𝐽t 𝑆))
1211sseq2d 3596 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴𝑆𝐴 (𝐽t 𝑆)))
1311difeq1d 3689 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐴) = ( (𝐽t 𝑆) ∖ 𝐴))
1413eleq1d 2672 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆)))
1512, 14anbi12d 743 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ (𝐴 (𝐽t 𝑆) ∧ ( (𝐽t 𝑆) ∖ 𝐴) ∈ (𝐽t 𝑆))))
16 elrest 15911 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)))
175, 16syldan 486 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)))
1817anbi2d 736 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ (𝐴𝑆 ∧ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆))))
192opncld 20647 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑜𝐽) → (𝑋𝑜) ∈ (Clsd‘𝐽))
2019adantlr 747 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑜𝐽) → (𝑋𝑜) ∈ (Clsd‘𝐽))
2120adantlr 747 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) → (𝑋𝑜) ∈ (Clsd‘𝐽))
2221adantr 480 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑋𝑜) ∈ (Clsd‘𝐽))
23 incom 3767 . . . . . . . . . . . . 13 (𝑋𝑆) = (𝑆𝑋)
24 df-ss 3554 . . . . . . . . . . . . . 14 (𝑆𝑋 ↔ (𝑆𝑋) = 𝑆)
2524biimpi 205 . . . . . . . . . . . . 13 (𝑆𝑋 → (𝑆𝑋) = 𝑆)
2623, 25syl5eq 2656 . . . . . . . . . . . 12 (𝑆𝑋 → (𝑋𝑆) = 𝑆)
2726ad4antlr 765 . . . . . . . . . . 11 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑋𝑆) = 𝑆)
2827difeq1d 3689 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → ((𝑋𝑆) ∖ 𝑜) = (𝑆𝑜))
29 difeq2 3684 . . . . . . . . . . . 12 ((𝑆𝐴) = (𝑜𝑆) → (𝑆 ∖ (𝑆𝐴)) = (𝑆 ∖ (𝑜𝑆)))
30 difindi 3840 . . . . . . . . . . . . 13 (𝑆 ∖ (𝑜𝑆)) = ((𝑆𝑜) ∪ (𝑆𝑆))
31 difid 3902 . . . . . . . . . . . . . 14 (𝑆𝑆) = ∅
3231uneq2i 3726 . . . . . . . . . . . . 13 ((𝑆𝑜) ∪ (𝑆𝑆)) = ((𝑆𝑜) ∪ ∅)
33 un0 3919 . . . . . . . . . . . . 13 ((𝑆𝑜) ∪ ∅) = (𝑆𝑜)
3430, 32, 333eqtri 2636 . . . . . . . . . . . 12 (𝑆 ∖ (𝑜𝑆)) = (𝑆𝑜)
3529, 34syl6eq 2660 . . . . . . . . . . 11 ((𝑆𝐴) = (𝑜𝑆) → (𝑆 ∖ (𝑆𝐴)) = (𝑆𝑜))
3635adantl 481 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑆 ∖ (𝑆𝐴)) = (𝑆𝑜))
37 dfss4 3820 . . . . . . . . . . . 12 (𝐴𝑆 ↔ (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3837biimpi 205 . . . . . . . . . . 11 (𝐴𝑆 → (𝑆 ∖ (𝑆𝐴)) = 𝐴)
3938ad3antlr 763 . . . . . . . . . 10 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → (𝑆 ∖ (𝑆𝐴)) = 𝐴)
4028, 36, 393eqtr2rd 2651 . . . . . . . . 9 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → 𝐴 = ((𝑋𝑆) ∖ 𝑜))
4123difeq1i 3686 . . . . . . . . . 10 ((𝑋𝑆) ∖ 𝑜) = ((𝑆𝑋) ∖ 𝑜)
42 indif2 3829 . . . . . . . . . 10 (𝑆 ∩ (𝑋𝑜)) = ((𝑆𝑋) ∖ 𝑜)
43 incom 3767 . . . . . . . . . 10 (𝑆 ∩ (𝑋𝑜)) = ((𝑋𝑜) ∩ 𝑆)
4441, 42, 433eqtr2i 2638 . . . . . . . . 9 ((𝑋𝑆) ∖ 𝑜) = ((𝑋𝑜) ∩ 𝑆)
4540, 44syl6eq 2660 . . . . . . . 8 (((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) ∧ 𝑜𝐽) ∧ (𝑆𝐴) = (𝑜𝑆)) → 𝐴 = ((𝑋𝑜) ∩ 𝑆))
46 ineq1 3769 . . . . . . . . . 10 (𝑥 = (𝑋𝑜) → (𝑥𝑆) = ((𝑋𝑜) ∩ 𝑆))
4746eqeq2d 2620 . . . . . . . . 9 (𝑥 = (𝑋𝑜) → (𝐴 = (𝑥𝑆) ↔ 𝐴 = ((𝑋𝑜) ∩ 𝑆)))
4847rspcev 3282 . . . . . . . 8 (((𝑋𝑜) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((𝑋𝑜) ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆))
4922, 45, 48syl2anc 691 . . . . . . 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 3013 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝐴𝑆) → (∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
5251expimpd 627 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ ∃𝑜𝐽 (𝑆𝐴) = (𝑜𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
5318, 52sylbid 229 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
54 difindi 3840 . . . . . . . . . 10 (𝑆 ∖ (𝑥𝑆)) = ((𝑆𝑥) ∪ (𝑆𝑆))
5531uneq2i 3726 . . . . . . . . . 10 ((𝑆𝑥) ∪ (𝑆𝑆)) = ((𝑆𝑥) ∪ ∅)
56 un0 3919 . . . . . . . . . 10 ((𝑆𝑥) ∪ ∅) = (𝑆𝑥)
5754, 55, 563eqtri 2636 . . . . . . . . 9 (𝑆 ∖ (𝑥𝑆)) = (𝑆𝑥)
58 difin2 3849 . . . . . . . . . 10 (𝑆𝑋 → (𝑆𝑥) = ((𝑋𝑥) ∩ 𝑆))
5958adantl 481 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝑥) = ((𝑋𝑥) ∩ 𝑆))
6057, 59syl5eq 2656 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∖ (𝑥𝑆)) = ((𝑋𝑥) ∩ 𝑆))
6160adantr 480 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥𝑆)) = ((𝑋𝑥) ∩ 𝑆))
62 simpll 786 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
635adantr 480 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑆 ∈ V)
642cldopn 20645 . . . . . . . . 9 (𝑥 ∈ (Clsd‘𝐽) → (𝑋𝑥) ∈ 𝐽)
6564adantl 481 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋𝑥) ∈ 𝐽)
66 elrestr 15912 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 ∈ V ∧ (𝑋𝑥) ∈ 𝐽) → ((𝑋𝑥) ∩ 𝑆) ∈ (𝐽t 𝑆))
6762, 63, 65, 66syl3anc 1318 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑋𝑥) ∩ 𝑆) ∈ (𝐽t 𝑆))
6861, 67eqeltrd 2688 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆))
69 inss2 3796 . . . . . 6 (𝑥𝑆) ⊆ 𝑆
7068, 69jctil 558 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆)))
71 sseq1 3589 . . . . . 6 (𝐴 = (𝑥𝑆) → (𝐴𝑆 ↔ (𝑥𝑆) ⊆ 𝑆))
72 difeq2 3684 . . . . . . 7 (𝐴 = (𝑥𝑆) → (𝑆𝐴) = (𝑆 ∖ (𝑥𝑆)))
7372eleq1d 2672 . . . . . 6 (𝐴 = (𝑥𝑆) → ((𝑆𝐴) ∈ (𝐽t 𝑆) ↔ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆)))
7471, 73anbi12d 743 . . . . 5 (𝐴 = (𝑥𝑆) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ ((𝑥𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥𝑆)) ∈ (𝐽t 𝑆))))
7570, 74syl5ibrcom 236 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐴 = (𝑥𝑆) → (𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆))))
7675rexlimdva 3013 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆) → (𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆))))
7753, 76impbid 201 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐴𝑆 ∧ (𝑆𝐴) ∈ (𝐽t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
7810, 15, 773bitr2d 295 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴 ∈ (Clsd‘(𝐽t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  Clsdccld 20630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633 This theorem is referenced by:  restcldi  20787  restcldr  20788  restcls  20795  consubclo  21037  cldllycmp  21108
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