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Theorem restcls 20895
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 1059 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ Top)
2 sstr 3591 . . . . . . . 8 ((𝑆𝑌𝑌𝑋) → 𝑆𝑋)
32ancoms 469 . . . . . . 7 ((𝑌𝑋𝑆𝑌) → 𝑆𝑋)
433adant1 1077 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑋)
5 restcls.1 . . . . . . 7 𝑋 = 𝐽
65clscld 20761 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
71, 4, 6syl2anc 692 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
8 eqid 2621 . . . . 5 (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)
9 ineq1 3785 . . . . . . 7 (𝑥 = ((cls‘𝐽)‘𝑆) → (𝑥𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
109eqeq2d 2631 . . . . . 6 (𝑥 = ((cls‘𝐽)‘𝑆) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)))
1110rspcev 3295 . . . . 5 ((((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌))
127, 8, 11sylancl 693 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌))
13 restcls.2 . . . . . . 7 𝐾 = (𝐽t 𝑌)
1413fveq2i 6151 . . . . . 6 (Clsd‘𝐾) = (Clsd‘(𝐽t 𝑌))
1514eleq2i 2690 . . . . 5 ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)))
165restcld 20886 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
17163adant3 1079 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
1815, 17syl5bb 272 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
1912, 18mpbird 247 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾))
205sscls 20770 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
211, 4, 20syl2anc 692 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
22 simp3 1061 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑌)
2321, 22ssind 3815 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
24 eqid 2621 . . . 4 𝐾 = 𝐾
2524clsss2 20786 . . 3 (((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ∧ 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
2619, 23, 25syl2anc 692 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
2713fveq2i 6151 . . . . . 6 (cls‘𝐾) = (cls‘(𝐽t 𝑌))
2827fveq1i 6149 . . . . 5 ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽t 𝑌))‘𝑆)
29 id 22 . . . . . . . . 9 (𝑌𝑋𝑌𝑋)
305topopn 20636 . . . . . . . . 9 (𝐽 ∈ Top → 𝑋𝐽)
31 ssexg 4764 . . . . . . . . 9 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
3229, 30, 31syl2anr 495 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
33 resttop 20874 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽t 𝑌) ∈ Top)
3432, 33syldan 487 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ Top)
35343adant3 1079 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝐽t 𝑌) ∈ Top)
365restuni 20876 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 = (𝐽t 𝑌))
37363adant3 1079 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 = (𝐽t 𝑌))
3822, 37sseqtrd 3620 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 (𝐽t 𝑌))
39 eqid 2621 . . . . . . 7 (𝐽t 𝑌) = (𝐽t 𝑌)
4039clscld 20761 . . . . . 6 (((𝐽t 𝑌) ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((cls‘(𝐽t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
4135, 38, 40syl2anc 692 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘(𝐽t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
4228, 41syl5eqel 2702 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
435restcld 20886 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌)))
44433adant3 1079 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌)))
4542, 44mpbid 222 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌))
4613, 34syl5eqel 2702 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝐾 ∈ Top)
47463adant3 1079 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ Top)
4813unieqi 4411 . . . . . . . . 9 𝐾 = (𝐽t 𝑌)
4948eqcomi 2630 . . . . . . . 8 (𝐽t 𝑌) = 𝐾
5049sscls 20770 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
5147, 38, 50syl2anc 692 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
5251adantr 481 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
53 inss1 3811 . . . . . . 7 (𝑥𝑌) ⊆ 𝑥
54 sseq1 3605 . . . . . . 7 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (((cls‘𝐾)‘𝑆) ⊆ 𝑥 ↔ (𝑥𝑌) ⊆ 𝑥))
5553, 54mpbiri 248 . . . . . 6 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
5655ad2antll 764 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
5752, 56sstrd 3593 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → 𝑆𝑥)
585clsss2 20786 . . . . . . . . . 10 ((𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
5958adantl 482 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
60 ssrin 3816 . . . . . . . . 9 (((cls‘𝐽)‘𝑆) ⊆ 𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥𝑌))
6159, 60syl 17 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥𝑌))
62 sseq2 3606 . . . . . . . 8 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥𝑌)))
6361, 62syl5ibrcom 237 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)))
6463expr 642 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆𝑥 → (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))))
6564com23 86 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (𝑆𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))))
6665impr 648 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → (𝑆𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)))
6757, 66mpd 15 . . 3 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))
6845, 67rexlimddv 3028 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))
6926, 68eqssd 3600 1 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  cin 3554  wss 3555   cuni 4402  cfv 5847  (class class class)co 6604  t crest 16002  Topctop 20617  Clsdccld 20730  clsccl 20732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733  df-cls 20735
This theorem is referenced by:  restlp  20897  resscdrg  23062
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