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Theorem innei 20910
 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 2620 . . . . 5 𝐽 = 𝐽
21neii1 20891 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
3 ssinss1 3833 . . . 4 (𝑁 𝐽 → (𝑁𝑀) ⊆ 𝐽)
42, 3syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ⊆ 𝐽)
543adant3 1079 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ⊆ 𝐽)
6 neii2 20893 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝐽 (𝑆𝑁))
7 neii2 20893 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑀))
86, 7anim12dan 881 . . . 4 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → (∃𝐽 (𝑆𝑁) ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑀)))
9 inopn 20685 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐽𝑣𝐽) → (𝑣) ∈ 𝐽)
1093expa 1263 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) → (𝑣) ∈ 𝐽)
11 ssin 3827 . . . . . . . . . . . . . 14 ((𝑆𝑆𝑣) ↔ 𝑆 ⊆ (𝑣))
1211biimpi 206 . . . . . . . . . . . . 13 ((𝑆𝑆𝑣) → 𝑆 ⊆ (𝑣))
13 ss2in 3832 . . . . . . . . . . . . 13 ((𝑁𝑣𝑀) → (𝑣) ⊆ (𝑁𝑀))
1412, 13anim12i 589 . . . . . . . . . . . 12 (((𝑆𝑆𝑣) ∧ (𝑁𝑣𝑀)) → (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀)))
1514an4s 868 . . . . . . . . . . 11 (((𝑆𝑁) ∧ (𝑆𝑣𝑣𝑀)) → (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀)))
16 sseq2 3619 . . . . . . . . . . . . 13 (𝑔 = (𝑣) → (𝑆𝑔𝑆 ⊆ (𝑣)))
17 sseq1 3618 . . . . . . . . . . . . 13 (𝑔 = (𝑣) → (𝑔 ⊆ (𝑁𝑀) ↔ (𝑣) ⊆ (𝑁𝑀)))
1816, 17anbi12d 746 . . . . . . . . . . . 12 (𝑔 = (𝑣) → ((𝑆𝑔𝑔 ⊆ (𝑁𝑀)) ↔ (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀))))
1918rspcev 3304 . . . . . . . . . . 11 (((𝑣) ∈ 𝐽 ∧ (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
2010, 15, 19syl2an 494 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) ∧ ((𝑆𝑁) ∧ (𝑆𝑣𝑣𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
2120expr 642 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) ∧ (𝑆𝑁)) → ((𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2221an32s 845 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐽) ∧ (𝑆𝑁)) ∧ 𝑣𝐽) → ((𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2322rexlimdva 3027 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐽) ∧ (𝑆𝑁)) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2423ex 450 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐽) → ((𝑆𝑁) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
2524rexlimdva 3027 . . . . 5 (𝐽 ∈ Top → (∃𝐽 (𝑆𝑁) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
2625imp32 449 . . . 4 ((𝐽 ∈ Top ∧ (∃𝐽 (𝑆𝑁) ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
278, 26syldan 487 . . 3 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
28273impb 1258 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
291neiss2 20886 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 𝐽)
301isnei 20888 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
3129, 30syldan 487 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
32313adant3 1079 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
335, 28, 32mpbir2and 956 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 196   ∧ wa 384   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988  ∃wrex 2910   ∩ cin 3566   ⊆ wss 3567  ∪ cuni 4427  ‘cfv 5876  Topctop 20679  neicnei 20882 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-top 20680  df-nei 20883 This theorem is referenced by:  neifil  21665  neificl  33520
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