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Theorem islly2 21197
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 21185 . . . 4 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
21adantl 482 . . 3 ((𝜑𝐽 ∈ Locally 𝐴) → 𝐽 ∈ Top)
3 simplr 791 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝐽 ∈ Locally 𝐴)
42adantr 481 . . . . . . 7 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝐽 ∈ Top)
5 islly2.2 . . . . . . . 8 𝑋 = 𝐽
65topopn 20636 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
74, 6syl 17 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝑋𝐽)
8 simpr 477 . . . . . 6 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → 𝑦𝑋)
9 llyi 21187 . . . . . 6 ((𝐽 ∈ Locally 𝐴𝑋𝐽𝑦𝑋) → ∃𝑢𝐽 (𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
103, 7, 8, 9syl3anc 1323 . . . . 5 (((𝜑𝐽 ∈ Locally 𝐴) ∧ 𝑦𝑋) → ∃𝑢𝐽 (𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
11 3simpc 1058 . . . . . 6 ((𝑢𝑋𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
1211reximi 3005 . . . . 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 2960 . . 3 ((𝜑𝐽 ∈ Locally 𝐴) → ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
152, 14jca 554 . 2 ((𝜑𝐽 ∈ Locally 𝐴) → (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
16 simprl 793 . . 3 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
17 elssuni 4433 . . . . . . . . 9 (𝑧𝐽𝑧 𝐽)
1817, 5syl6sseqr 3631 . . . . . . . 8 (𝑧𝐽𝑧𝑋)
1918adantl 482 . . . . . . 7 (((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) → 𝑧𝑋)
20 ssralv 3645 . . . . . . 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 798 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
23 simplrl 799 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑧𝐽)
24 simprl 793 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝐽)
25 inopn 20629 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑧𝐽𝑢𝐽) → (𝑧𝑢) ∈ 𝐽)
2622, 23, 24, 25syl3anc 1323 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ 𝐽)
27 inss1 3811 . . . . . . . . . . . . 13 (𝑧𝑢) ⊆ 𝑧
28 vex 3189 . . . . . . . . . . . . . 14 𝑧 ∈ V
2928elpw2 4788 . . . . . . . . . . . . 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 3776 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ (𝐽 ∩ 𝒫 𝑧))
33 simplrr 800 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑧)
34 simprrl 803 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑢)
3533, 34elind 3776 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦 ∈ (𝑧𝑢))
36 inss2 3812 . . . . . . . . . . . . 13 (𝑧𝑢) ⊆ 𝑢
3736a1i 11 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ⊆ 𝑢)
38 restabs 20879 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑧𝑢) ⊆ 𝑢𝑢𝐽) → ((𝐽t 𝑢) ↾t (𝑧𝑢)) = (𝐽t (𝑧𝑢)))
3922, 37, 24, 38syl3anc 1323 . . . . . . . . . . 11 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ((𝐽t 𝑢) ↾t (𝑧𝑢)) = (𝐽t (𝑧𝑢)))
40 elrestr 16010 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑧𝐽) → (𝑧𝑢) ∈ (𝐽t 𝑢))
4122, 24, 23, 40syl3anc 1323 . . . . . . . . . . . 12 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑧𝑢) ∈ (𝐽t 𝑢))
42 simprrr 804 . . . . . . . . . . . . 13 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t 𝑢) ∈ 𝐴)
43 restlly.1 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
4443ralrimivva 2965 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑗𝐴𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴)
4544ad3antrrr 765 . . . . . . . . . . . . 13 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∀𝑗𝐴𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴)
46 oveq1 6611 . . . . . . . . . . . . . . . 16 (𝑗 = (𝐽t 𝑢) → (𝑗t 𝑥) = ((𝐽t 𝑢) ↾t 𝑥))
4746eleq1d 2683 . . . . . . . . . . . . . . 15 (𝑗 = (𝐽t 𝑢) → ((𝑗t 𝑥) ∈ 𝐴 ↔ ((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴))
4847raleqbi1dv 3135 . . . . . . . . . . . . . 14 (𝑗 = (𝐽t 𝑢) → (∀𝑥𝑗 (𝑗t 𝑥) ∈ 𝐴 ↔ ∀𝑥 ∈ (𝐽t 𝑢)((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴))
4948rspcv 3291 . . . . . . . . . . . . 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 6612 . . . . . . . . . . . . . 14 (𝑥 = (𝑧𝑢) → ((𝐽t 𝑢) ↾t 𝑥) = ((𝐽t 𝑢) ↾t (𝑧𝑢)))
5251eleq1d 2683 . . . . . . . . . . . . 13 (𝑥 = (𝑧𝑢) → (((𝐽t 𝑢) ↾t 𝑥) ∈ 𝐴 ↔ ((𝐽t 𝑢) ↾t (𝑧𝑢)) ∈ 𝐴))
5352rspcv 3291 . . . . . . . . . . . 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 2699 . . . . . . . . . 10 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t (𝑧𝑢)) ∈ 𝐴)
56 eleq2 2687 . . . . . . . . . . . 12 (𝑣 = (𝑧𝑢) → (𝑦𝑣𝑦 ∈ (𝑧𝑢)))
57 oveq2 6612 . . . . . . . . . . . . 13 (𝑣 = (𝑧𝑢) → (𝐽t 𝑣) = (𝐽t (𝑧𝑢)))
5857eleq1d 2683 . . . . . . . . . . . 12 (𝑣 = (𝑧𝑢) → ((𝐽t 𝑣) ∈ 𝐴 ↔ (𝐽t (𝑧𝑢)) ∈ 𝐴))
5956, 58anbi12d 746 . . . . . . . . . . 11 (𝑣 = (𝑧𝑢) → ((𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴) ↔ (𝑦 ∈ (𝑧𝑢) ∧ (𝐽t (𝑧𝑢)) ∈ 𝐴)))
6059rspcev 3295 . . . . . . . . . 10 (((𝑧𝑢) ∈ (𝐽 ∩ 𝒫 𝑧) ∧ (𝑦 ∈ (𝑧𝑢) ∧ (𝐽t (𝑧𝑢)) ∈ 𝐴)) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
6132, 35, 55, 60syl12anc 1321 . . . . . . . . 9 ((((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
6261rexlimdvaa 3025 . . . . . . . 8 (((𝜑𝐽 ∈ Top) ∧ (𝑧𝐽𝑦𝑧)) → (∃𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6362anassrs 679 . . . . . . 7 ((((𝜑𝐽 ∈ Top) ∧ 𝑧𝐽) ∧ 𝑦𝑧) → (∃𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∃𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6463ralimdva 2956 . . . . . 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 2963 . . . 4 ((𝜑𝐽 ∈ Top) → (∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6766impr 648 . . 3 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴))
68 islly 21181 . . 3 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑧𝐽𝑦𝑧𝑣 ∈ (𝐽 ∩ 𝒫 𝑧)(𝑦𝑣 ∧ (𝐽t 𝑣) ∈ 𝐴)))
6916, 67, 68sylanbrc 697 . 2 ((𝜑 ∧ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
7015, 69impbida 876 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402  (class class class)co 6604  t crest 16002  Topctop 20617  Locally clly 21177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-rest 16004  df-top 20621  df-lly 21179
This theorem is referenced by: (None)
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