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Theorem islly2 23508
Description: An alternative expression for 𝐽 ∈ Locally 𝐴 when 𝐴 passes to open subspaces: A space is locally 𝐴 if every point is contained in an open neighborhood with property 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
Hypotheses
Ref Expression
restlly.1 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
islly2.2 𝑋 = 𝐽
Assertion
Ref Expression
islly2 (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
Distinct variable groups:   𝑢,𝑗,𝑥,𝑦,𝐴   𝑗,𝐽,𝑢,𝑥,𝑦   𝜑,𝑗,𝑢,𝑥,𝑦   𝑢,𝑋,𝑦
Allowed substitution hints:   𝑋(𝑥,𝑗)

Proof of Theorem islly2
Dummy variables 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 23496 . . . 4 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
21adantl 481 . . 3 ((𝜑𝐽 ∈ Locally 𝐴) → 𝐽 ∈ Top)
3 simplr 769 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝐽 ∈ Locally 𝐴)
42adantr 480 . . . . . . 7 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝐽 ∈ Top)
5 islly2.2 . . . . . . . 8 𝑋 = 𝐽
65topopn 22928 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
74, 6syl 17 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝑋𝐽)
8 simpr 484 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝑦𝑋)
9 llyi 23498 . . . . . 6 ((𝐽 ∈ Locally 𝐴𝑋𝐽𝑦𝑋) → ∃𝑢𝐽 (𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
103, 7, 8, 9syl3anc 1370 . . . . 5 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → ∃𝑢𝐽 (𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
11 3simpc 1149 . . . . . 6 ((𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
1211reximi 3082 . . . . 5 (∃𝑢𝐽 (𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∃𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
1310, 12syl 17 . . . 4 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → ∃𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
1413ralrimiva 3144 . . 3 ((𝜑𝐽 ∈ Locally 𝐴) → ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
152, 14jca 511 . 2 ((𝜑𝐽 ∈ Locally 𝐴) → (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
16 simprl 771 . . 3 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
17 elssuni 4942 . . . . . . . . 9 (𝑧𝐽𝑧 𝐽)
1817, 5sseqtrrdi 4047 . . . . . . . 8 (𝑧𝐽𝑧𝑋)
1918adantl 481 . . . . . . 7 (((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) → 𝑧𝑋)
20 ssralv 4064 . . . . . . 7 (𝑧𝑋 → (∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∀𝑦𝑧𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
2119, 20syl 17 . . . . . 6 (((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) → (∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∀𝑦𝑧𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
22 simpllr 776 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
23 simplrl 777 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑧𝐽)
24 simprl 771 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝐽)
25 inopn 22921 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑧𝐽𝑢𝐽) → (𝑧𝑢) ∈ 𝐽)
2622, 23, 24, 25syl3anc 1370 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ 𝐽)
27 vex 3482 . . . . . . . . . . . . 13 𝑧 ∈ V
28 inss1 4245 . . . . . . . . . . . . 13 (𝑧𝑢) ⊆ 𝑧
2927, 28elpwi2 5341 . . . . . . . . . . . 12 (𝑧𝑢) ∈ 𝒫 𝑧
3029a1i 11 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ 𝒫 𝑧)
3126, 30elind 4210 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ (𝐽 ∩ 𝒫 𝑧))
32 simplrr 778 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑧)
33 simprrl 781 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑢)
3432, 33elind 4210 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦 ∈ (𝑧𝑢))
35 inss2 4246 . . . . . . . . . . . . 13 (𝑧𝑢) ⊆ 𝑢
3635a1i 11 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ⊆ 𝑢)
37 restabs 23189 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑧𝑢) ⊆ 𝑢𝑢𝐽) → ((𝐽t 𝑢) ↾t (𝑧𝑢)) = (𝐽t (𝑧𝑢)))
3822, 36, 24, 37syl3anc 1370 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ((𝐽t 𝑢) ↾t (𝑧𝑢)) = (𝐽t (𝑧𝑢)))
39 oveq2 7439 . . . . . . . . . . . . 13 (𝑥 = (𝑧𝑢) → ((𝐽t 𝑢) ↾t 𝑥) = ((𝐽t 𝑢) ↾t (𝑧𝑢)))
4039eleq1d 2824 . . . . . . . . . . . 12 (𝑥 = (𝑧𝑢) → (((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽t 𝑢) ↾t (𝑧𝑢)) ∈ 𝐴))
41 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑗 = (𝐽t 𝑢) → (𝑗t 𝑥) = ((𝐽t 𝑢) ↾t 𝑥))
4241eleq1d 2824 . . . . . . . . . . . . . 14 (𝑗 = (𝐽t 𝑢) → ((𝑗t 𝑥) ∈ 𝐴 ↔ ((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴))
4342raleqbi1dv 3336 . . . . . . . . . . . . 13 (𝑗 = (𝐽t 𝑢) → (∀𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴 ↔ ∀𝑥 ∈ (𝐽t 𝑢)((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴))
44 restlly.1 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
4544ralrimivva 3200 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑗𝐴𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴)
4645ad3antrrr 730 . . . . . . . . . . . . 13 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∀𝑗𝐴𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴)
47 simprrr 782 . . . . . . . . . . . . 13 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t 𝑢) ∈ 𝐴)
4843, 46, 47rspcdva 3623 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∀𝑥 ∈ (𝐽t 𝑢)((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴)
49 elrestr 17475 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑧𝐽) → (𝑧𝑢) ∈ (𝐽t 𝑢))
5022, 24, 23, 49syl3anc 1370 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ (𝐽t 𝑢))
5140, 48, 50rspcdva 3623 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ((𝐽t 𝑢) ↾t (𝑧𝑢)) ∈ 𝐴)
5238, 51eqeltrrd 2840 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t (𝑧𝑢)) ∈ 𝐴)
53 eleq2 2828 . . . . . . . . . . . 12 (𝑣 = (𝑧𝑢) → (𝑦𝑣𝑦 ∈ (𝑧𝑢)))
54 oveq2 7439 . . . . . . . . . . . . 13 (𝑣 = (𝑧𝑢) → (𝐽t 𝑣) = (𝐽t (𝑧𝑢)))
5554eleq1d 2824 . . . . . . . . . . . 12 (𝑣 = (𝑧𝑢) → ((𝐽t 𝑣) ∈ 𝐴 ↔ (𝐽t (𝑧𝑢)) ∈ 𝐴))
5653, 55anbi12d 632 . . . . . . . . . . 11 (𝑣 = (𝑧𝑢) → ((𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴) ↔ (𝑦 ∈ (𝑧𝑢) ∧ (𝐽t (𝑧𝑢)) ∈ 𝐴)))
5756rspcev 3622 . . . . . . . . . 10 (((𝑧𝑢) ∈ (𝐽 ∩ 𝒫 𝑧) ∧ (𝑦 ∈ (𝑧𝑢) ∧ (𝐽t (𝑧𝑢)) ∈ 𝐴)) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
5831, 34, 52, 57syl12anc 837 . . . . . . . . 9 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
5958rexlimdvaa 3154 . . . . . . . 8 (((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) → (∃𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6059anassrs 467 . . . . . . 7 ((((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) ∧ 𝑦𝑧) → (∃𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6160ralimdva 3165 . . . . . 6 (((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) → (∀𝑦𝑧𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∀𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6221, 61syld 47 . . . . 5 (((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) → (∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∀𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6362ralrimdva 3152 . . . 4 ((𝜑𝐽 ∈ Top) → (∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6463impr 454 . . 3 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
65 islly 23492 . . 3 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6616, 64, 65sylanbrc 583 . 2 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
6715, 66impbida 801 1 (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912  (class class class)co 7431  t crest 17467  Topctop 22915  Locally clly 23488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17469  df-top 22916  df-lly 23490
This theorem is referenced by: (None)
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