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Theorem llyrest 23493
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 23480 . . 3 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
2 resttop 23168 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
31, 2sylan 580 . 2 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
4 restopn2 23185 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
51, 4sylan 580 . . . 4 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
6 simp1l 1198 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝐽 ∈ Locally 𝐴)
7 simp2l 1200 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑥𝐽)
8 simp3 1139 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑦𝑥)
9 llyi 23482 . . . . . . . . 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 1222 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
13 simpl2r 1228 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑥𝐵)
1412, 13sstrd 3994 . . . . . . . . . . . . 13 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝐵)
156, 1syl 17 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝐽 ∈ Top)
1615adantr 480 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝐽 ∈ Top)
17 simpl1r 1226 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝐵𝐽)
18 restopn2 23185 . . . . . . . . . . . . . 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 713 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝐽t 𝐵))
21 velpw 4605 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
2212, 21sylibr 234 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
2320, 22elind 4200 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥))
24 simprr2 1223 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑦𝑣)
25 restabs 23173 . . . . . . . . . . . . 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 1224 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝐽t 𝑣) ∈ 𝐴)
2826, 27eqeltrd 2841 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)
2923, 24, 28jca32 515 . . . . . . . . . 10 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
3029ex 412 . . . . . . . . 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 1119 . . . . . 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 412 . . . 4 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → ((𝑥𝐽𝑥𝐵) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
365, 35sylbid 240 . . 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 23476 . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600  (class class class)co 7431  t crest 17465  Topctop 22899  Locally clly 23472
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-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-lly 23474
This theorem is referenced by:  loclly  23495  llyidm  23496
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