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Theorem llyrest 21098
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 21085 . . 3 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
2 resttop 20774 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
31, 2sylan 487 . 2 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Top)
4 restopn2 20791 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
51, 4sylan 487 . . . 4 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) ↔ (𝑥𝐽𝑥𝐵)))
6 simp1l 1078 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝐽 ∈ Locally 𝐴)
7 simp2l 1080 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑥𝐽)
8 simp3 1056 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → 𝑦𝑥)
9 llyi 21087 . . . . . . . . 9 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑣𝐽 (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
106, 7, 8, 9syl3anc 1318 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ∃𝑣𝐽 (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
11 simprl 790 . . . . . . . . . . . . 13 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝐽)
12 simprr1 1102 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
13 simpl2r 1108 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑥𝐵)
1412, 13sstrd 3578 . . . . . . . . . . . . 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 1106 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝐵𝐽)
18 restopn2 20791 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑣 ∈ (𝐽t 𝐵) ↔ (𝑣𝐽𝑣𝐵)))
1916, 17, 18syl2anc 691 . . . . . . . . . . . . 13 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (𝐽t 𝐵) ↔ (𝑣𝐽𝑣𝐵)))
2011, 14, 19mpbir2and 959 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝐽t 𝐵))
21 selpw 4115 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
2212, 21sylibr 223 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
2320, 22elind 3760 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥))
24 simprr2 1103 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → 𝑦𝑣)
25 restabs 20779 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑣𝐵𝐵𝐽) → ((𝐽t 𝐵) ↾t 𝑣) = (𝐽t 𝑣))
2616, 14, 17, 25syl3anc 1318 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑣) = (𝐽t 𝑣))
27 simprr3 1104 . . . . . . . . . . . 12 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝐽t 𝑣) ∈ 𝐴)
2826, 27eqeltrd 2688 . . . . . . . . . . 11 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)
2923, 24, 28jca32 556 . . . . . . . . . 10 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) ∧ (𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))) → (𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
3029ex 449 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵) ∧ 𝑦𝑥) → ((𝑣𝐽 ∧ (𝑣𝑥𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)) → (𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))))
3130reximdv2 2997 . . . . . . . 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 1257 . . . . . 6 ((((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) ∧ 𝑦𝑥) → ∃𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
3433ralrimiva 2949 . . . . 5 (((𝐽 ∈ Locally 𝐴𝐵𝐽) ∧ (𝑥𝐽𝑥𝐵)) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
3534ex 449 . . . 4 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → ((𝑥𝐽𝑥𝐵) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
365, 35sylbid 229 . . 3 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝑥 ∈ (𝐽t 𝐵) → ∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴)))
3736ralrimiv 2948 . 2 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → ∀𝑥 ∈ (𝐽t 𝐵)∀𝑦𝑥𝑣 ∈ ((𝐽t 𝐵) ∩ 𝒫 𝑥)(𝑦𝑣 ∧ ((𝐽t 𝐵) ↾t 𝑣) ∈ 𝐴))
38 islly 21081 . 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 695 1 ((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  cin 3539  wss 3540  𝒫 cpw 4108  (class class class)co 6549  t crest 15904  Topctop 20517  Locally clly 21077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-lly 21079
This theorem is referenced by:  loclly  21100  llyidm  21101
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