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Theorem restlly 21196
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 3583 . . . 4 ((𝜑𝑗𝐴) → 𝑗 ∈ Top)
3 simprl 793 . . . . . . 7 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥𝑗)
4 vex 3189 . . . . . . . . 9 𝑥 ∈ V
54pwid 4145 . . . . . . . 8 𝑥 ∈ 𝒫 𝑥
65a1i 11 . . . . . . 7 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥 ∈ 𝒫 𝑥)
73, 6elind 3776 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑥 ∈ (𝑗 ∩ 𝒫 𝑥))
8 simprr 795 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → 𝑦𝑥)
9 restlly.1 . . . . . . . 8 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
109anassrs 679 . . . . . . 7 (((𝜑𝑗𝐴) ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ 𝐴)
1110adantrr 752 . . . . . 6 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → (𝑗t 𝑥) ∈ 𝐴)
12 elequ2 2001 . . . . . . . 8 (𝑢 = 𝑥 → (𝑦𝑢𝑦𝑥))
13 oveq2 6612 . . . . . . . . 9 (𝑢 = 𝑥 → (𝑗t 𝑢) = (𝑗t 𝑥))
1413eleq1d 2683 . . . . . . . 8 (𝑢 = 𝑥 → ((𝑗t 𝑢) ∈ 𝐴 ↔ (𝑗t 𝑥) ∈ 𝐴))
1512, 14anbi12d 746 . . . . . . 7 (𝑢 = 𝑥 → ((𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ (𝑦𝑥 ∧ (𝑗t 𝑥) ∈ 𝐴)))
1615rspcev 3295 . . . . . 6 ((𝑥 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦𝑥 ∧ (𝑗t 𝑥) ∈ 𝐴)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
177, 8, 11, 16syl12anc 1321 . . . . 5 (((𝜑𝑗𝐴) ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
1817ralrimivva 2965 . . . 4 ((𝜑𝑗𝐴) → ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴))
19 islly 21181 . . . 4 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)))
202, 18, 19sylanbrc 697 . . 3 ((𝜑𝑗𝐴) → 𝑗 ∈ Locally 𝐴)
2120ex 450 . 2 (𝜑 → (𝑗𝐴𝑗 ∈ Locally 𝐴))
2221ssrdv 3589 1 (𝜑𝐴 ⊆ Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130  (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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
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-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  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-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-lly 21179
This theorem is referenced by:  llyidm  21201  nllyidm  21202  toplly  21203  hauslly  21205  lly1stc  21209
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