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Theorem innei 12405
 Description: The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property Vii of [BourbakiTop1] p. I.3 for binary intersections. (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 2140 . . . . 5 𝐽 = 𝐽
21neii1 12389 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
3 ssinss1 3311 . . . 4 (𝑁 𝐽 → (𝑁𝑀) ⊆ 𝐽)
42, 3syl 14 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ⊆ 𝐽)
543adant3 1002 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ⊆ 𝐽)
6 neii2 12391 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝐽 (𝑆𝑁))
7 neii2 12391 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑀))
86, 7anim12dan 590 . . . 4 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → (∃𝐽 (𝑆𝑁) ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑀)))
9 inopn 12243 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐽𝑣𝐽) → (𝑣) ∈ 𝐽)
1093expa 1182 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) → (𝑣) ∈ 𝐽)
11 ssin 3304 . . . . . . . . . . . . 13 ((𝑆𝑆𝑣) ↔ 𝑆 ⊆ (𝑣))
1211biimpi 119 . . . . . . . . . . . 12 ((𝑆𝑆𝑣) → 𝑆 ⊆ (𝑣))
13 ss2in 3310 . . . . . . . . . . . 12 ((𝑁𝑣𝑀) → (𝑣) ⊆ (𝑁𝑀))
1412, 13anim12i 336 . . . . . . . . . . 11 (((𝑆𝑆𝑣) ∧ (𝑁𝑣𝑀)) → (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀)))
1514an4s 578 . . . . . . . . . 10 (((𝑆𝑁) ∧ (𝑆𝑣𝑣𝑀)) → (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀)))
16 sseq2 3127 . . . . . . . . . . . 12 (𝑔 = (𝑣) → (𝑆𝑔𝑆 ⊆ (𝑣)))
17 sseq1 3126 . . . . . . . . . . . 12 (𝑔 = (𝑣) → (𝑔 ⊆ (𝑁𝑀) ↔ (𝑣) ⊆ (𝑁𝑀)))
1816, 17anbi12d 465 . . . . . . . . . . 11 (𝑔 = (𝑣) → ((𝑆𝑔𝑔 ⊆ (𝑁𝑀)) ↔ (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀))))
1918rspcev 2794 . . . . . . . . . 10 (((𝑣) ∈ 𝐽 ∧ (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
2010, 15, 19syl2an 287 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) ∧ ((𝑆𝑁) ∧ (𝑆𝑣𝑣𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
2120expr 373 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) ∧ (𝑆𝑁)) → ((𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2221an32s 558 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐽) ∧ (𝑆𝑁)) ∧ 𝑣𝐽) → ((𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2322rexlimdva 2553 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐽) ∧ (𝑆𝑁)) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2423rexlimdva2 2556 . . . . 5 (𝐽 ∈ Top → (∃𝐽 (𝑆𝑁) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
2524imp32 255 . . . 4 ((𝐽 ∈ Top ∧ (∃𝐽 (𝑆𝑁) ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
268, 25syldan 280 . . 3 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
27263impb 1178 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
281neiss2 12384 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 𝐽)
291isnei 12386 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
3028, 29syldan 280 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
31303adant3 1002 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
325, 27, 31mpbir2and 929 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∃wrex 2418   ∩ cin 3076   ⊆ wss 3077  ∪ cuni 3745  ‘cfv 5134  Topctop 12237  neicnei 12380 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-pow 4107  ax-pr 4141 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4225  df-xp 4556  df-rel 4557  df-cnv 4558  df-co 4559  df-dm 4560  df-rn 4561  df-res 4562  df-ima 4563  df-iota 5099  df-fun 5136  df-fn 5137  df-f 5138  df-f1 5139  df-fo 5140  df-f1o 5141  df-fv 5142  df-top 12238  df-nei 12381 This theorem is referenced by: (None)
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