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Theorem restlly 21488
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 3744 . . . 4 ((𝜑𝑗𝐴) → 𝑗 ∈ Top)
3 simprl 811 . . . . . . 7 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥𝑗)
4 vex 3343 . . . . . . . . 9 𝑥 ∈ V
54pwid 4318 . . . . . . . 8 𝑥 ∈ 𝒫 𝑥
65a1i 11 . . . . . . 7 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥 ∈ 𝒫 𝑥)
73, 6elind 3941 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥 ∈ (𝑗 ∩ 𝒫 𝑥))
8 simprr 813 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑦𝑥)
9 restlly.1 . . . . . . . 8 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
109anassrs 683 . . . . . . 7 (((𝜑𝑗𝐴) ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ 𝐴)
1110adantrr 755 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → (𝑗t 𝑥) ∈ 𝐴)
12 elequ2 2153 . . . . . . . 8 (𝑢 = 𝑥 → (𝑦𝑢𝑦𝑥))
13 oveq2 6821 . . . . . . . . 9 (𝑢 = 𝑥 → (𝑗t 𝑢) = (𝑗t 𝑥))
1413eleq1d 2824 . . . . . . . 8 (𝑢 = 𝑥 → ((𝑗t 𝑢) ∈ 𝐴 ↔ (𝑗t 𝑥) ∈ 𝐴))
1512, 14anbi12d 749 . . . . . . 7 (𝑢 = 𝑥 → ((𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ (𝑦𝑥 ∧ (𝑗t 𝑥) ∈ 𝐴)))
1615rspcev 3449 . . . . . 6 ((𝑥 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦𝑥 ∧ (𝑗t 𝑥) ∈ 𝐴)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
177, 8, 11, 16syl12anc 1475 . . . . 5 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
1817ralrimivva 3109 . . . 4 ((𝜑𝑗𝐴) → ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
19 islly 21473 . . . 4 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)))
202, 18, 19sylanbrc 701 . . 3 ((𝜑𝑗𝐴) → 𝑗 ∈ Locally 𝐴)
2120ex 449 . 2 (𝜑 → (𝑗𝐴𝑗 ∈ Locally 𝐴))
2221ssrdv 3750 1 (𝜑𝐴 ⊆ Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wral 3050  wrex 3051  cin 3714  wss 3715  𝒫 cpw 4302  (class class class)co 6813  t crest 16283  Topctop 20900  Locally clly 21469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816  df-lly 21471
This theorem is referenced by:  llyidm  21493  nllyidm  21494  toplly  21495  hauslly  21497  lly1stc  21501
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