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Theorem restlly 21096
 Description: If the property 𝐴 passes to open subspaces, then a space which is 𝐴 is also locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
Hypotheses
Ref Expression
restlly.1 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
restlly.2 (𝜑𝐴 ⊆ Top)
Assertion
Ref Expression
restlly (𝜑𝐴 ⊆ Locally 𝐴)
Distinct variable groups:   𝑥,𝑗,𝐴   𝜑,𝑗,𝑥

Proof of Theorem restlly
Dummy variables 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 restlly.2 . . . . 5 (𝜑𝐴 ⊆ Top)
21sselda 3568 . . . 4 ((𝜑𝑗𝐴) → 𝑗 ∈ Top)
3 simprl 790 . . . . . . 7 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥𝑗)
4 vex 3176 . . . . . . . . 9 𝑥 ∈ V
54pwid 4122 . . . . . . . 8 𝑥 ∈ 𝒫 𝑥
65a1i 11 . . . . . . 7 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥 ∈ 𝒫 𝑥)
73, 6elind 3760 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥 ∈ (𝑗 ∩ 𝒫 𝑥))
8 simprr 792 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑦𝑥)
9 restlly.1 . . . . . . . 8 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
109anassrs 678 . . . . . . 7 (((𝜑𝑗𝐴) ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ 𝐴)
1110adantrr 749 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → (𝑗t 𝑥) ∈ 𝐴)
12 eleq2 2677 . . . . . . . 8 (𝑢 = 𝑥 → (𝑦𝑢𝑦𝑥))
13 oveq2 6557 . . . . . . . . 9 (𝑢 = 𝑥 → (𝑗t 𝑢) = (𝑗t 𝑥))
1413eleq1d 2672 . . . . . . . 8 (𝑢 = 𝑥 → ((𝑗t 𝑢) ∈ 𝐴 ↔ (𝑗t 𝑥) ∈ 𝐴))
1512, 14anbi12d 743 . . . . . . 7 (𝑢 = 𝑥 → ((𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ (𝑦𝑥 ∧ (𝑗t 𝑥) ∈ 𝐴)))
1615rspcev 3282 . . . . . 6 ((𝑥 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦𝑥 ∧ (𝑗t 𝑥) ∈ 𝐴)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
177, 8, 11, 16syl12anc 1316 . . . . 5 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
1817ralrimivva 2954 . . . 4 ((𝜑𝑗𝐴) → ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
19 islly 21081 . . . 4 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)))
202, 18, 19sylanbrc 695 . . 3 ((𝜑𝑗𝐴) → 𝑗 ∈ Locally 𝐴)
2120ex 449 . 2 (𝜑 → (𝑗𝐴𝑗 ∈ Locally 𝐴))
2221ssrdv 3574 1 (𝜑𝐴 ⊆ Locally 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-lly 21079 This theorem is referenced by:  llyidm  21101  nllyidm  21102  toplly  21103  hauslly  21105  lly1stc  21109
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