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Theorem llyrest 23209
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 23196 . . 3 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
2 resttop 22884 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
31, 2sylan 580 . 2 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
4 restopn2 22901 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
51, 4sylan 580 . . . 4 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
6 simp1l 1197 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝐽 ∈ Locally 𝐴)
7 simp2l 1199 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑥𝐽)
8 simp3 1138 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑦𝑥)
9 llyi 23198 . . . . . . . . 9 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑣𝐽 (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
106, 7, 8, 9syl3anc 1371 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ∃𝑣𝐽 (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
11 simprl 769 . . . . . . . . . . . . 13 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝐽)
12 simprr1 1221 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
13 simpl2r 1227 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑥𝐵)
1412, 13sstrd 3992 . . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝐵𝐽)
18 restopn2 22901 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑣 ∈ (𝐽t 𝐵) ↔ (𝑣𝐽𝑣𝐵)))
1916, 17, 18syl2anc 584 . . . . . . . . . . . . 13 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (𝐽t 𝐵) ↔ (𝑣𝐽𝑣𝐵)))
2011, 14, 19mpbir2and 711 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝐽t 𝐵))
21 velpw 4607 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
2212, 21sylibr 233 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
2320, 22elind 4194 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥))
24 simprr2 1222 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑦𝑣)
25 restabs 22889 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑣𝐵𝐵𝐽) → ((𝐽t 𝐵) ↾t 𝑣) = (𝐽t 𝑣))
2616, 14, 17, 25syl3anc 1371 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑣) = (𝐽t 𝑣))
27 simprr3 1223 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝐽t 𝑣) ∈ 𝐴)
2826, 27eqeltrd 2833 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)
2923, 24, 28jca32 516 . . . . . . . . . 10 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
3029ex 413 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ((𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)) → (𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))))
3130reximdv2 3164 . . . . . . . 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 1118 . . . . . 6 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) ∧ 𝑦𝑥) → ∃𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
3433ralrimiva 3146 . . . . 5 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
3534ex 413 . . . 4 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → ((𝑥𝐽𝑥𝐵) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
365, 35sylbid 239 . . 3 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
3736ralrimiv 3145 . 2 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → ∀𝑥 ∈ (𝐽t 𝐵)∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
38 islly 23192 . 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 583 1 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070  cin 3947  wss 3948  𝒫 cpw 4602  (class class class)co 7411  t crest 17370  Topctop 22615  Locally clly 23188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17372  df-topgen 17393  df-top 22616  df-topon 22633  df-bases 22669  df-lly 23190
This theorem is referenced by:  loclly  23211  llyidm  23212
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