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Theorem llyrest 22382
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 22369 . . 3 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
2 resttop 22057 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
31, 2sylan 583 . 2 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
4 restopn2 22074 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
51, 4sylan 583 . . . 4 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
6 simp1l 1199 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝐽 ∈ Locally 𝐴)
7 simp2l 1201 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑥𝐽)
8 simp3 1140 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑦𝑥)
9 llyi 22371 . . . . . . . . 9 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑣𝐽 (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
106, 7, 8, 9syl3anc 1373 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ∃𝑣𝐽 (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
11 simprl 771 . . . . . . . . . . . . 13 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝐽)
12 simprr1 1223 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
13 simpl2r 1229 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑥𝐵)
1412, 13sstrd 3911 . . . . . . . . . . . . 13 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝐵)
156, 1syl 17 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝐽 ∈ Top)
1615adantr 484 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝐽 ∈ Top)
17 simpl1r 1227 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝐵𝐽)
18 restopn2 22074 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑣 ∈ (𝐽t 𝐵) ↔ (𝑣𝐽𝑣𝐵)))
1916, 17, 18syl2anc 587 . . . . . . . . . . . . 13 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (𝐽t 𝐵) ↔ (𝑣𝐽𝑣𝐵)))
2011, 14, 19mpbir2and 713 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝐽t 𝐵))
21 velpw 4518 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
2212, 21sylibr 237 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
2320, 22elind 4108 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥))
24 simprr2 1224 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑦𝑣)
25 restabs 22062 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑣𝐵𝐵𝐽) → ((𝐽t 𝐵) ↾t 𝑣) = (𝐽t 𝑣))
2616, 14, 17, 25syl3anc 1373 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑣) = (𝐽t 𝑣))
27 simprr3 1225 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝐽t 𝑣) ∈ 𝐴)
2826, 27eqeltrd 2838 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)
2923, 24, 28jca32 519 . . . . . . . . . 10 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
3029ex 416 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ((𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)) → (𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))))
3130reximdv2 3190 . . . . . . . 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 1120 . . . . . 6 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) ∧ 𝑦𝑥) → ∃𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
3433ralrimiva 3105 . . . . 5 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
3534ex 416 . . . 4 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → ((𝑥𝐽𝑥𝐵) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
365, 35sylbid 243 . . 3 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
3736ralrimiv 3104 . 2 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → ∀𝑥 ∈ (𝐽t 𝐵)∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
38 islly 22365 . 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 586 1 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  cin 3865  wss 3866  𝒫 cpw 4513  (class class class)co 7213  t crest 16925  Topctop 21790  Locally clly 22361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-en 8627  df-fin 8630  df-fi 9027  df-rest 16927  df-topgen 16948  df-top 21791  df-topon 21808  df-bases 21843  df-lly 22363
This theorem is referenced by:  loclly  22384  llyidm  22385
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