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Theorem restlly 21500
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 3752 . . . 4 ((𝜑𝑗𝐴) → 𝑗 ∈ Top)
3 simprl 754 . . . . . . 7 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥𝑗)
4 vex 3354 . . . . . . . . 9 𝑥 ∈ V
54pwid 4313 . . . . . . . 8 𝑥 ∈ 𝒫 𝑥
65a1i 11 . . . . . . 7 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥 ∈ 𝒫 𝑥)
73, 6elind 3949 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥 ∈ (𝑗 ∩ 𝒫 𝑥))
8 simprr 756 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑦𝑥)
9 restlly.1 . . . . . . . 8 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
109anassrs 453 . . . . . . 7 (((𝜑𝑗𝐴) ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ 𝐴)
1110adantrr 696 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → (𝑗t 𝑥) ∈ 𝐴)
12 elequ2 2159 . . . . . . . 8 (𝑢 = 𝑥 → (𝑦𝑢𝑦𝑥))
13 oveq2 6799 . . . . . . . . 9 (𝑢 = 𝑥 → (𝑗t 𝑢) = (𝑗t 𝑥))
1413eleq1d 2835 . . . . . . . 8 (𝑢 = 𝑥 → ((𝑗t 𝑢) ∈ 𝐴 ↔ (𝑗t 𝑥) ∈ 𝐴))
1512, 14anbi12d 616 . . . . . . 7 (𝑢 = 𝑥 → ((𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ (𝑦𝑥 ∧ (𝑗t 𝑥) ∈ 𝐴)))
1615rspcev 3460 . . . . . 6 ((𝑥 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦𝑥 ∧ (𝑗t 𝑥) ∈ 𝐴)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
177, 8, 11, 16syl12anc 1474 . . . . 5 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
1817ralrimivva 3120 . . . 4 ((𝜑𝑗𝐴) → ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
19 islly 21485 . . . 4 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)))
202, 18, 19sylanbrc 572 . . 3 ((𝜑𝑗𝐴) → 𝑗 ∈ Locally 𝐴)
2120ex 397 . 2 (𝜑 → (𝑗𝐴𝑗 ∈ Locally 𝐴))
2221ssrdv 3758 1 (𝜑𝐴 ⊆ Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  wral 3061  wrex 3062  cin 3722  wss 3723  𝒫 cpw 4297  (class class class)co 6791  t crest 16282  Topctop 20911  Locally clly 21481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5992  df-fv 6037  df-ov 6794  df-lly 21483
This theorem is referenced by:  llyidm  21505  nllyidm  21506  toplly  21507  hauslly  21509  lly1stc  21513
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