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Theorem restlly 23380
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 3978 . . . 4 ((𝜑𝑗𝐴) → 𝑗 ∈ Top)
3 simprl 770 . . . . . . 7 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥𝑗)
4 vex 3474 . . . . . . . . 9 𝑥 ∈ V
54pwid 4620 . . . . . . . 8 𝑥 ∈ 𝒫 𝑥
65a1i 11 . . . . . . 7 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥 ∈ 𝒫 𝑥)
73, 6elind 4190 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥 ∈ (𝑗 ∩ 𝒫 𝑥))
8 simprr 772 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑦𝑥)
9 restlly.1 . . . . . . . 8 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
109anassrs 467 . . . . . . 7 (((𝜑𝑗𝐴) ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ 𝐴)
1110adantrr 716 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → (𝑗t 𝑥) ∈ 𝐴)
12 elequ2 2114 . . . . . . . 8 (𝑢 = 𝑥 → (𝑦𝑢𝑦𝑥))
13 oveq2 7422 . . . . . . . . 9 (𝑢 = 𝑥 → (𝑗t 𝑢) = (𝑗t 𝑥))
1413eleq1d 2814 . . . . . . . 8 (𝑢 = 𝑥 → ((𝑗t 𝑢) ∈ 𝐴 ↔ (𝑗t 𝑥) ∈ 𝐴))
1512, 14anbi12d 631 . . . . . . 7 (𝑢 = 𝑥 → ((𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ (𝑦𝑥 ∧ (𝑗t 𝑥) ∈ 𝐴)))
1615rspcev 3608 . . . . . 6 ((𝑥 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦𝑥 ∧ (𝑗t 𝑥) ∈ 𝐴)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
177, 8, 11, 16syl12anc 836 . . . . 5 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
1817ralrimivva 3196 . . . 4 ((𝜑𝑗𝐴) → ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
19 islly 23365 . . . 4 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)))
202, 18, 19sylanbrc 582 . . 3 ((𝜑𝑗𝐴) → 𝑗 ∈ Locally 𝐴)
2120ex 412 . 2 (𝜑 → (𝑗𝐴𝑗 ∈ Locally 𝐴))
2221ssrdv 3984 1 (𝜑𝐴 ⊆ Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  wral 3057  wrex 3066  cin 3944  wss 3945  𝒫 cpw 4598  (class class class)co 7414  t crest 17395  Topctop 22788  Locally clly 23361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-lly 23363
This theorem is referenced by:  llyidm  23385  nllyidm  23386  toplly  23387  hauslly  23389  lly1stc  23393
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