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Theorem llyrest 21198
Description: An open subspace of a locally 𝐴 space is also locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llyrest ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Locally 𝐴)

Proof of Theorem llyrest
Dummy variables 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 21185 . . 3 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
2 resttop 20874 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
31, 2sylan 488 . 2 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
4 restopn2 20891 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
51, 4sylan 488 . . . 4 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
6 simp1l 1083 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝐽 ∈ Locally 𝐴)
7 simp2l 1085 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑥𝐽)
8 simp3 1061 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑦𝑥)
9 llyi 21187 . . . . . . . . 9 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑣𝐽 (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
106, 7, 8, 9syl3anc 1323 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ∃𝑣𝐽 (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
11 simprl 793 . . . . . . . . . . . . 13 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝐽)
12 simprr1 1107 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
13 simpl2r 1113 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑥𝐵)
1412, 13sstrd 3593 . . . . . . . . . . . . 13 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝐵)
156, 1syl 17 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝐽 ∈ Top)
1615adantr 481 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝐽 ∈ Top)
17 simpl1r 1111 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝐵𝐽)
18 restopn2 20891 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑣 ∈ (𝐽t 𝐵) ↔ (𝑣𝐽𝑣𝐵)))
1916, 17, 18syl2anc 692 . . . . . . . . . . . . 13 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (𝐽t 𝐵) ↔ (𝑣𝐽𝑣𝐵)))
2011, 14, 19mpbir2and 956 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝐽t 𝐵))
21 selpw 4137 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
2212, 21sylibr 224 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
2320, 22elind 3776 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥))
24 simprr2 1108 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑦𝑣)
25 restabs 20879 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑣𝐵𝐵𝐽) → ((𝐽t 𝐵) ↾t 𝑣) = (𝐽t 𝑣))
2616, 14, 17, 25syl3anc 1323 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑣) = (𝐽t 𝑣))
27 simprr3 1109 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝐽t 𝑣) ∈ 𝐴)
2826, 27eqeltrd 2698 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)
2923, 24, 28jca32 557 . . . . . . . . . 10 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
3029ex 450 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ((𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)) → (𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))))
3130reximdv2 3008 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → (∃𝑣𝐽 (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴) → ∃𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
3210, 31mpd 15 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ∃𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
33323expa 1262 . . . . . 6 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) ∧ 𝑦𝑥) → ∃𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
3433ralrimiva 2960 . . . . 5 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
3534ex 450 . . . 4 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → ((𝑥𝐽𝑥𝐵) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
365, 35sylbid 230 . . 3 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
3736ralrimiv 2959 . 2 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → ∀𝑥 ∈ (𝐽t 𝐵)∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
38 islly 21181 . 2 ((𝐽t 𝐵) ∈ Locally 𝐴 ↔ ((𝐽t 𝐵) ∈ Top ∧ ∀𝑥 ∈ (𝐽t 𝐵)∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
393, 37, 38sylanbrc 697 1 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130  (class class class)co 6604  t crest 16002  Topctop 20617  Locally clly 21177
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-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-lly 21179
This theorem is referenced by:  loclly  21200  llyidm  21201
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