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Theorem restcls 23176
Description: A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
Hypotheses
Ref Expression
restcls.1 𝑋 = 𝐽
restcls.2 𝐾 = (𝐽t 𝑌)
Assertion
Ref Expression
restcls ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))

Proof of Theorem restcls
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp1 1133 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ Top)
2 sstr 3988 . . . . . . . 8 ((𝑆𝑌𝑌𝑋) → 𝑆𝑋)
32ancoms 457 . . . . . . 7 ((𝑌𝑋𝑆𝑌) → 𝑆𝑋)
433adant1 1127 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑋)
5 restcls.1 . . . . . . 7 𝑋 = 𝐽
65clscld 23042 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
71, 4, 6syl2anc 582 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
8 eqid 2726 . . . . 5 (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)
9 ineq1 4206 . . . . . 6 (𝑥 = ((cls‘𝐽)‘𝑆) → (𝑥𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
109rspceeqv 3630 . . . . 5 ((((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌))
117, 8, 10sylancl 584 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌))
12 restcls.2 . . . . . . 7 𝐾 = (𝐽t 𝑌)
1312fveq2i 6904 . . . . . 6 (Clsd‘𝐾) = (Clsd‘(𝐽t 𝑌))
1413eleq2i 2818 . . . . 5 ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)))
155restcld 23167 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
16153adant3 1129 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
1714, 16bitrid 282 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
1811, 17mpbird 256 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾))
195sscls 23051 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
201, 4, 19syl2anc 582 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
21 simp3 1135 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑌)
2220, 21ssind 4234 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
23 eqid 2726 . . . 4 𝐾 = 𝐾
2423clsss2 23067 . . 3 (((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ∧ 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
2518, 22, 24syl2anc 582 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
2612fveq2i 6904 . . . . . 6 (cls‘𝐾) = (cls‘(𝐽t 𝑌))
2726fveq1i 6902 . . . . 5 ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽t 𝑌))‘𝑆)
28 id 22 . . . . . . . . 9 (𝑌𝑋𝑌𝑋)
295topopn 22899 . . . . . . . . 9 (𝐽 ∈ Top → 𝑋𝐽)
30 ssexg 5328 . . . . . . . . 9 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
3128, 29, 30syl2anr 595 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
32 resttop 23155 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽t 𝑌) ∈ Top)
3331, 32syldan 589 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ Top)
34333adant3 1129 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝐽t 𝑌) ∈ Top)
355restuni 23157 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 = (𝐽t 𝑌))
36353adant3 1129 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 = (𝐽t 𝑌))
3721, 36sseqtrd 4020 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 (𝐽t 𝑌))
38 eqid 2726 . . . . . . 7 (𝐽t 𝑌) = (𝐽t 𝑌)
3938clscld 23042 . . . . . 6 (((𝐽t 𝑌) ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((cls‘(𝐽t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
4034, 37, 39syl2anc 582 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘(𝐽t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
4127, 40eqeltrid 2830 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
425restcld 23167 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌)))
43423adant3 1129 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌)))
4441, 43mpbid 231 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌))
4512, 33eqeltrid 2830 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝐾 ∈ Top)
46453adant3 1129 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ Top)
4712unieqi 4925 . . . . . . . . 9 𝐾 = (𝐽t 𝑌)
4847eqcomi 2735 . . . . . . . 8 (𝐽t 𝑌) = 𝐾
4948sscls 23051 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
5046, 37, 49syl2anc 582 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
5150adantr 479 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
52 inss1 4230 . . . . . . 7 (𝑥𝑌) ⊆ 𝑥
53 sseq1 4005 . . . . . . 7 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (((cls‘𝐾)‘𝑆) ⊆ 𝑥 ↔ (𝑥𝑌) ⊆ 𝑥))
5452, 53mpbiri 257 . . . . . 6 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
5554ad2antll 727 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
5651, 55sstrd 3990 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → 𝑆𝑥)
575clsss2 23067 . . . . . . . . . 10 ((𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
5857adantl 480 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
5958ssrind 4237 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥𝑌))
60 sseq2 4006 . . . . . . . 8 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥𝑌)))
6159, 60syl5ibrcom 246 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)))
6261expr 455 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆𝑥 → (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))))
6362com23 86 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (𝑆𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))))
6463impr 453 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → (𝑆𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)))
6556, 64mpd 15 . . 3 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))
6644, 65rexlimddv 3151 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))
6725, 66eqssd 3997 1 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wrex 3060  Vcvv 3462  cin 3946  wss 3947   cuni 4913  cfv 6554  (class class class)co 7424  t crest 17435  Topctop 22886  Clsdccld 23011  clsccl 23013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-en 8975  df-fin 8978  df-fi 9454  df-rest 17437  df-topgen 17458  df-top 22887  df-topon 22904  df-bases 22940  df-cld 23014  df-cls 23016
This theorem is referenced by:  restlp  23178  resscdrg  25377  restcls2  48247
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