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Theorem islly2 21487
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 21475 . . . 4 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
21adantl 473 . . 3 ((𝜑𝐽 ∈ Locally 𝐴) → 𝐽 ∈ Top)
3 simplr 809 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝐽 ∈ Locally 𝐴)
42adantr 472 . . . . . . 7 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝐽 ∈ Top)
5 islly2.2 . . . . . . . 8 𝑋 = 𝐽
65topopn 20911 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
74, 6syl 17 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝑋𝐽)
8 simpr 479 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝑦𝑋)
9 llyi 21477 . . . . . 6 ((𝐽 ∈ Locally 𝐴𝑋𝐽𝑦𝑋) → ∃𝑢𝐽 (𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
103, 7, 8, 9syl3anc 1477 . . . . 5 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → ∃𝑢𝐽 (𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
11 3simpc 1147 . . . . . 6 ((𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
1211reximi 3147 . . . . 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 3102 . . 3 ((𝜑𝐽 ∈ Locally 𝐴) → ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
152, 14jca 555 . 2 ((𝜑𝐽 ∈ Locally 𝐴) → (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
16 simprl 811 . . 3 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
17 elssuni 4617 . . . . . . . . 9 (𝑧𝐽𝑧 𝐽)
1817, 5syl6sseqr 3791 . . . . . . . 8 (𝑧𝐽𝑧𝑋)
1918adantl 473 . . . . . . 7 (((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) → 𝑧𝑋)
20 ssralv 3805 . . . . . . 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 817 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
23 simplrl 819 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑧𝐽)
24 simprl 811 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝐽)
25 inopn 20904 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑧𝐽𝑢𝐽) → (𝑧𝑢) ∈ 𝐽)
2622, 23, 24, 25syl3anc 1477 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ 𝐽)
27 inss1 3974 . . . . . . . . . . . . 13 (𝑧𝑢) ⊆ 𝑧
28 vex 3341 . . . . . . . . . . . . . 14 𝑧 ∈ V
2928elpw2 4975 . . . . . . . . . . . . 13 ((𝑧𝑢) ∈ 𝒫 𝑧 ↔ (𝑧𝑢) ⊆ 𝑧)
3027, 29mpbir 221 . . . . . . . . . . . 12 (𝑧𝑢) ∈ 𝒫 𝑧
3130a1i 11 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ 𝒫 𝑧)
3226, 31elind 3939 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ (𝐽 ∩ 𝒫 𝑧))
33 simplrr 820 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑧)
34 simprrl 823 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑢)
3533, 34elind 3939 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦 ∈ (𝑧𝑢))
36 inss2 3975 . . . . . . . . . . . . 13 (𝑧𝑢) ⊆ 𝑢
3736a1i 11 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ⊆ 𝑢)
38 restabs 21169 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑧𝑢) ⊆ 𝑢𝑢𝐽) → ((𝐽t 𝑢) ↾t (𝑧𝑢)) = (𝐽t (𝑧𝑢)))
3922, 37, 24, 38syl3anc 1477 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ((𝐽t 𝑢) ↾t (𝑧𝑢)) = (𝐽t (𝑧𝑢)))
40 elrestr 16289 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑧𝐽) → (𝑧𝑢) ∈ (𝐽t 𝑢))
4122, 24, 23, 40syl3anc 1477 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ (𝐽t 𝑢))
42 simprrr 824 . . . . . . . . . . . . 13 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t 𝑢) ∈ 𝐴)
43 restlly.1 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
4443ralrimivva 3107 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑗𝐴𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴)
4544ad3antrrr 768 . . . . . . . . . . . . 13 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∀𝑗𝐴𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴)
46 oveq1 6818 . . . . . . . . . . . . . . . 16 (𝑗 = (𝐽t 𝑢) → (𝑗t 𝑥) = ((𝐽t 𝑢) ↾t 𝑥))
4746eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑗 = (𝐽t 𝑢) → ((𝑗t 𝑥) ∈ 𝐴 ↔ ((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴))
4847raleqbi1dv 3283 . . . . . . . . . . . . . 14 (𝑗 = (𝐽t 𝑢) → (∀𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴 ↔ ∀𝑥 ∈ (𝐽t 𝑢)((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴))
4948rspcv 3443 . . . . . . . . . . . . 13 ((𝐽t 𝑢) ∈ 𝐴 → (∀𝑗𝐴𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴 → ∀𝑥 ∈ (𝐽t 𝑢)((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴))
5042, 45, 49sylc 65 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∀𝑥 ∈ (𝐽t 𝑢)((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴)
51 oveq2 6819 . . . . . . . . . . . . . 14 (𝑥 = (𝑧𝑢) → ((𝐽t 𝑢) ↾t 𝑥) = ((𝐽t 𝑢) ↾t (𝑧𝑢)))
5251eleq1d 2822 . . . . . . . . . . . . 13 (𝑥 = (𝑧𝑢) → (((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽t 𝑢) ↾t (𝑧𝑢)) ∈ 𝐴))
5352rspcv 3443 . . . . . . . . . . . 12 ((𝑧𝑢) ∈ (𝐽t 𝑢) → (∀𝑥 ∈ (𝐽t 𝑢)((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴 → ((𝐽t 𝑢) ↾t (𝑧𝑢)) ∈ 𝐴))
5441, 50, 53sylc 65 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ((𝐽t 𝑢) ↾t (𝑧𝑢)) ∈ 𝐴)
5539, 54eqeltrrd 2838 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t (𝑧𝑢)) ∈ 𝐴)
56 eleq2 2826 . . . . . . . . . . . 12 (𝑣 = (𝑧𝑢) → (𝑦𝑣𝑦 ∈ (𝑧𝑢)))
57 oveq2 6819 . . . . . . . . . . . . 13 (𝑣 = (𝑧𝑢) → (𝐽t 𝑣) = (𝐽t (𝑧𝑢)))
5857eleq1d 2822 . . . . . . . . . . . 12 (𝑣 = (𝑧𝑢) → ((𝐽t 𝑣) ∈ 𝐴 ↔ (𝐽t (𝑧𝑢)) ∈ 𝐴))
5956, 58anbi12d 749 . . . . . . . . . . 11 (𝑣 = (𝑧𝑢) → ((𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴) ↔ (𝑦 ∈ (𝑧𝑢) ∧ (𝐽t (𝑧𝑢)) ∈ 𝐴)))
6059rspcev 3447 . . . . . . . . . 10 (((𝑧𝑢) ∈ (𝐽 ∩ 𝒫 𝑧) ∧ (𝑦 ∈ (𝑧𝑢) ∧ (𝐽t (𝑧𝑢)) ∈ 𝐴)) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
6132, 35, 55, 60syl12anc 1475 . . . . . . . . 9 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
6261rexlimdvaa 3168 . . . . . . . 8 (((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) → (∃𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6362anassrs 683 . . . . . . 7 ((((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) ∧ 𝑦𝑧) → (∃𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6463ralimdva 3098 . . . . . 6 (((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) → (∀𝑦𝑧𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∀𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6521, 64syld 47 . . . . 5 (((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) → (∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∀𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6665ralrimdva 3105 . . . 4 ((𝜑𝐽 ∈ Top) → (∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6766impr 650 . . 3 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
68 islly 21471 . . 3 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6916, 67, 68sylanbrc 701 . 2 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
7015, 69impbida 913 1 (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1630  wcel 2137  wral 3048  wrex 3049  cin 3712  wss 3713  𝒫 cpw 4300   cuni 4586  (class class class)co 6811  t crest 16281  Topctop 20898  Locally clly 21467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-rest 16283  df-top 20899  df-lly 21469
This theorem is referenced by: (None)
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