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Theorem restcls 23189
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 1137 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ Top)
2 sstr 3992 . . . . . . . 8 ((𝑆𝑌𝑌𝑋) → 𝑆𝑋)
32ancoms 458 . . . . . . 7 ((𝑌𝑋𝑆𝑌) → 𝑆𝑋)
433adant1 1131 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑋)
5 restcls.1 . . . . . . 7 𝑋 = 𝐽
65clscld 23055 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
71, 4, 6syl2anc 584 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
8 eqid 2737 . . . . 5 (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)
9 ineq1 4213 . . . . . 6 (𝑥 = ((cls‘𝐽)‘𝑆) → (𝑥𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
109rspceeqv 3645 . . . . 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 586 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌))
12 restcls.2 . . . . . . 7 𝐾 = (𝐽t 𝑌)
1312fveq2i 6909 . . . . . 6 (Clsd‘𝐾) = (Clsd‘(𝐽t 𝑌))
1413eleq2i 2833 . . . . 5 ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)))
155restcld 23180 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
16153adant3 1133 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
1714, 16bitrid 283 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
1811, 17mpbird 257 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾))
195sscls 23064 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
201, 4, 19syl2anc 584 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
21 simp3 1139 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑌)
2220, 21ssind 4241 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
23 eqid 2737 . . . 4 𝐾 = 𝐾
2423clsss2 23080 . . 3 (((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ∧ 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
2518, 22, 24syl2anc 584 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
2612fveq2i 6909 . . . . . 6 (cls‘𝐾) = (cls‘(𝐽t 𝑌))
2726fveq1i 6907 . . . . 5 ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽t 𝑌))‘𝑆)
28 id 22 . . . . . . . . 9 (𝑌𝑋𝑌𝑋)
295topopn 22912 . . . . . . . . 9 (𝐽 ∈ Top → 𝑋𝐽)
30 ssexg 5323 . . . . . . . . 9 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
3128, 29, 30syl2anr 597 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
32 resttop 23168 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽t 𝑌) ∈ Top)
3331, 32syldan 591 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ Top)
34333adant3 1133 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝐽t 𝑌) ∈ Top)
355restuni 23170 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 = (𝐽t 𝑌))
36353adant3 1133 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 = (𝐽t 𝑌))
3721, 36sseqtrd 4020 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 (𝐽t 𝑌))
38 eqid 2737 . . . . . . 7 (𝐽t 𝑌) = (𝐽t 𝑌)
3938clscld 23055 . . . . . 6 (((𝐽t 𝑌) ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((cls‘(𝐽t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
4034, 37, 39syl2anc 584 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘(𝐽t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
4127, 40eqeltrid 2845 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
425restcld 23180 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌)))
43423adant3 1133 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌)))
4441, 43mpbid 232 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌))
4512, 33eqeltrid 2845 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝐾 ∈ Top)
46453adant3 1133 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ Top)
4712unieqi 4919 . . . . . . . . 9 𝐾 = (𝐽t 𝑌)
4847eqcomi 2746 . . . . . . . 8 (𝐽t 𝑌) = 𝐾
4948sscls 23064 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
5046, 37, 49syl2anc 584 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
5150adantr 480 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
52 inss1 4237 . . . . . . 7 (𝑥𝑌) ⊆ 𝑥
53 sseq1 4009 . . . . . . 7 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (((cls‘𝐾)‘𝑆) ⊆ 𝑥 ↔ (𝑥𝑌) ⊆ 𝑥))
5452, 53mpbiri 258 . . . . . 6 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
5554ad2antll 729 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
5651, 55sstrd 3994 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → 𝑆𝑥)
575clsss2 23080 . . . . . . . . . 10 ((𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
5857adantl 481 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
5958ssrind 4244 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥𝑌))
60 sseq2 4010 . . . . . . . 8 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥𝑌)))
6159, 60syl5ibrcom 247 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)))
6261expr 456 . . . . . 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 454 . . . 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 3161 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))
6725, 66eqssd 4001 1 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cin 3950  wss 3951   cuni 4907  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  Clsdccld 23024  clsccl 23026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-cls 23029
This theorem is referenced by:  restlp  23191  resscdrg  25392  restcls2  48811
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