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Theorem islly2 22851
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 22839 . . . 4 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
21adantl 483 . . 3 ((𝜑𝐽 ∈ Locally 𝐴) → 𝐽 ∈ Top)
3 simplr 768 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝐽 ∈ Locally 𝐴)
42adantr 482 . . . . . . 7 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝐽 ∈ Top)
5 islly2.2 . . . . . . . 8 𝑋 = 𝐽
65topopn 22271 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
74, 6syl 17 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝑋𝐽)
8 simpr 486 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝑦𝑋)
9 llyi 22841 . . . . . 6 ((𝐽 ∈ Locally 𝐴𝑋𝐽𝑦𝑋) → ∃𝑢𝐽 (𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
103, 7, 8, 9syl3anc 1372 . . . . 5 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → ∃𝑢𝐽 (𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
11 3simpc 1151 . . . . . 6 ((𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
1211reximi 3084 . . . . 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 3140 . . 3 ((𝜑𝐽 ∈ Locally 𝐴) → ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
152, 14jca 513 . 2 ((𝜑𝐽 ∈ Locally 𝐴) → (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
16 simprl 770 . . 3 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
17 elssuni 4899 . . . . . . . . 9 (𝑧𝐽𝑧 𝐽)
1817, 5sseqtrrdi 3996 . . . . . . . 8 (𝑧𝐽𝑧𝑋)
1918adantl 483 . . . . . . 7 (((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) → 𝑧𝑋)
20 ssralv 4011 . . . . . . 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 775 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
23 simplrl 776 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑧𝐽)
24 simprl 770 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝐽)
25 inopn 22264 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑧𝐽𝑢𝐽) → (𝑧𝑢) ∈ 𝐽)
2622, 23, 24, 25syl3anc 1372 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ 𝐽)
27 vex 3448 . . . . . . . . . . . . 13 𝑧 ∈ V
28 inss1 4189 . . . . . . . . . . . . 13 (𝑧𝑢) ⊆ 𝑧
2927, 28elpwi2 5304 . . . . . . . . . . . 12 (𝑧𝑢) ∈ 𝒫 𝑧
3029a1i 11 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ 𝒫 𝑧)
3126, 30elind 4155 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ (𝐽 ∩ 𝒫 𝑧))
32 simplrr 777 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑧)
33 simprrl 780 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑢)
3432, 33elind 4155 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦 ∈ (𝑧𝑢))
35 inss2 4190 . . . . . . . . . . . . 13 (𝑧𝑢) ⊆ 𝑢
3635a1i 11 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ⊆ 𝑢)
37 restabs 22532 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑧𝑢) ⊆ 𝑢𝑢𝐽) → ((𝐽t 𝑢) ↾t (𝑧𝑢)) = (𝐽t (𝑧𝑢)))
3822, 36, 24, 37syl3anc 1372 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ((𝐽t 𝑢) ↾t (𝑧𝑢)) = (𝐽t (𝑧𝑢)))
39 oveq2 7366 . . . . . . . . . . . . 13 (𝑥 = (𝑧𝑢) → ((𝐽t 𝑢) ↾t 𝑥) = ((𝐽t 𝑢) ↾t (𝑧𝑢)))
4039eleq1d 2819 . . . . . . . . . . . 12 (𝑥 = (𝑧𝑢) → (((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽t 𝑢) ↾t (𝑧𝑢)) ∈ 𝐴))
41 oveq1 7365 . . . . . . . . . . . . . . 15 (𝑗 = (𝐽t 𝑢) → (𝑗t 𝑥) = ((𝐽t 𝑢) ↾t 𝑥))
4241eleq1d 2819 . . . . . . . . . . . . . 14 (𝑗 = (𝐽t 𝑢) → ((𝑗t 𝑥) ∈ 𝐴 ↔ ((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴))
4342raleqbi1dv 3306 . . . . . . . . . . . . 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 3194 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑗𝐴𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴)
4645ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∀𝑗𝐴𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴)
47 simprrr 781 . . . . . . . . . . . . 13 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t 𝑢) ∈ 𝐴)
4843, 46, 47rspcdva 3581 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∀𝑥 ∈ (𝐽t 𝑢)((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴)
49 elrestr 17315 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑧𝐽) → (𝑧𝑢) ∈ (𝐽t 𝑢))
5022, 24, 23, 49syl3anc 1372 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ (𝐽t 𝑢))
5140, 48, 50rspcdva 3581 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ((𝐽t 𝑢) ↾t (𝑧𝑢)) ∈ 𝐴)
5238, 51eqeltrrd 2835 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t (𝑧𝑢)) ∈ 𝐴)
53 eleq2 2823 . . . . . . . . . . . 12 (𝑣 = (𝑧𝑢) → (𝑦𝑣𝑦 ∈ (𝑧𝑢)))
54 oveq2 7366 . . . . . . . . . . . . 13 (𝑣 = (𝑧𝑢) → (𝐽t 𝑣) = (𝐽t (𝑧𝑢)))
5554eleq1d 2819 . . . . . . . . . . . 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 3580 . . . . . . . . . 10 (((𝑧𝑢) ∈ (𝐽 ∩ 𝒫 𝑧) ∧ (𝑦 ∈ (𝑧𝑢) ∧ (𝐽t (𝑧𝑢)) ∈ 𝐴)) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
5831, 34, 52, 57syl12anc 836 . . . . . . . . 9 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
5958rexlimdvaa 3150 . . . . . . . 8 (((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) → (∃𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6059anassrs 469 . . . . . . 7 ((((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) ∧ 𝑦𝑧) → (∃𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6160ralimdva 3161 . . . . . 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 3148 . . . 4 ((𝜑𝐽 ∈ Top) → (∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6463impr 456 . . 3 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
65 islly 22835 . . 3 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6616, 64, 65sylanbrc 584 . 2 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
6715, 66impbida 800 1 (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  Vcvv 3444  cin 3910  wss 3911  𝒫 cpw 4561   cuni 4866  (class class class)co 7358  t crest 17307  Topctop 22258  Locally clly 22831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-rest 17309  df-top 22259  df-lly 22833
This theorem is referenced by: (None)
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