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Theorem innei 20677
Description: The intersection of two neighborhoods of a set is also a neighborhood of the set. Proposition Vii of [BourbakiTop1] p. I.3 . (Contributed by FL, 28-Sep-2006.)
Assertion
Ref Expression
innei ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆))

Proof of Theorem innei
Dummy variables 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2605 . . . . 5 𝐽 = 𝐽
21neii1 20658 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
3 ssinss1 3798 . . . 4 (𝑁 𝐽 → (𝑁𝑀) ⊆ 𝐽)
42, 3syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ⊆ 𝐽)
543adant3 1073 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ⊆ 𝐽)
6 neii2 20660 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝐽 (𝑆𝑁))
7 neii2 20660 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑀))
86, 7anim12dan 877 . . . 4 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → (∃𝐽 (𝑆𝑁) ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑀)))
9 inopn 20467 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐽𝑣𝐽) → (𝑣) ∈ 𝐽)
1093expa 1256 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) → (𝑣) ∈ 𝐽)
11 ssin 3792 . . . . . . . . . . . . . 14 ((𝑆𝑆𝑣) ↔ 𝑆 ⊆ (𝑣))
1211biimpi 204 . . . . . . . . . . . . 13 ((𝑆𝑆𝑣) → 𝑆 ⊆ (𝑣))
13 ss2in 3797 . . . . . . . . . . . . 13 ((𝑁𝑣𝑀) → (𝑣) ⊆ (𝑁𝑀))
1412, 13anim12i 587 . . . . . . . . . . . 12 (((𝑆𝑆𝑣) ∧ (𝑁𝑣𝑀)) → (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀)))
1514an4s 864 . . . . . . . . . . 11 (((𝑆𝑁) ∧ (𝑆𝑣𝑣𝑀)) → (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀)))
16 sseq2 3585 . . . . . . . . . . . . 13 (𝑔 = (𝑣) → (𝑆𝑔𝑆 ⊆ (𝑣)))
17 sseq1 3584 . . . . . . . . . . . . 13 (𝑔 = (𝑣) → (𝑔 ⊆ (𝑁𝑀) ↔ (𝑣) ⊆ (𝑁𝑀)))
1816, 17anbi12d 742 . . . . . . . . . . . 12 (𝑔 = (𝑣) → ((𝑆𝑔𝑔 ⊆ (𝑁𝑀)) ↔ (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀))))
1918rspcev 3277 . . . . . . . . . . 11 (((𝑣) ∈ 𝐽 ∧ (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
2010, 15, 19syl2an 492 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) ∧ ((𝑆𝑁) ∧ (𝑆𝑣𝑣𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
2120expr 640 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) ∧ (𝑆𝑁)) → ((𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2221an32s 841 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐽) ∧ (𝑆𝑁)) ∧ 𝑣𝐽) → ((𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2322rexlimdva 3008 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐽) ∧ (𝑆𝑁)) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2423ex 448 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐽) → ((𝑆𝑁) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
2524rexlimdva 3008 . . . . 5 (𝐽 ∈ Top → (∃𝐽 (𝑆𝑁) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
2625imp32 447 . . . 4 ((𝐽 ∈ Top ∧ (∃𝐽 (𝑆𝑁) ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
278, 26syldan 485 . . 3 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
28273impb 1251 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
291neiss2 20653 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 𝐽)
301isnei 20655 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
3129, 30syldan 485 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
32313adant3 1073 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
335, 28, 32mpbir2and 958 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wrex 2892  cin 3534  wss 3535   cuni 4362  cfv 5786  Topctop 20455  neicnei 20649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-top 20459  df-nei 20650
This theorem is referenced by:  neifil  21432  neificl  32518
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