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Theorem innei 21131
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 2760 . . . . 5 𝐽 = 𝐽
21neii1 21112 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
3 ssinss1 3984 . . . 4 (𝑁 𝐽 → (𝑁𝑀) ⊆ 𝐽)
42, 3syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ⊆ 𝐽)
543adant3 1127 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ⊆ 𝐽)
6 neii2 21114 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝐽 (𝑆𝑁))
7 neii2 21114 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑀))
86, 7anim12dan 918 . . . 4 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → (∃𝐽 (𝑆𝑁) ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑀)))
9 inopn 20906 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐽𝑣𝐽) → (𝑣) ∈ 𝐽)
1093expa 1112 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) → (𝑣) ∈ 𝐽)
11 ssin 3978 . . . . . . . . . . . . . 14 ((𝑆𝑆𝑣) ↔ 𝑆 ⊆ (𝑣))
1211biimpi 206 . . . . . . . . . . . . 13 ((𝑆𝑆𝑣) → 𝑆 ⊆ (𝑣))
13 ss2in 3983 . . . . . . . . . . . . 13 ((𝑁𝑣𝑀) → (𝑣) ⊆ (𝑁𝑀))
1412, 13anim12i 591 . . . . . . . . . . . 12 (((𝑆𝑆𝑣) ∧ (𝑁𝑣𝑀)) → (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀)))
1514an4s 904 . . . . . . . . . . 11 (((𝑆𝑁) ∧ (𝑆𝑣𝑣𝑀)) → (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀)))
16 sseq2 3768 . . . . . . . . . . . . 13 (𝑔 = (𝑣) → (𝑆𝑔𝑆 ⊆ (𝑣)))
17 sseq1 3767 . . . . . . . . . . . . 13 (𝑔 = (𝑣) → (𝑔 ⊆ (𝑁𝑀) ↔ (𝑣) ⊆ (𝑁𝑀)))
1816, 17anbi12d 749 . . . . . . . . . . . 12 (𝑔 = (𝑣) → ((𝑆𝑔𝑔 ⊆ (𝑁𝑀)) ↔ (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀))))
1918rspcev 3449 . . . . . . . . . . 11 (((𝑣) ∈ 𝐽 ∧ (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
2010, 15, 19syl2an 495 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) ∧ ((𝑆𝑁) ∧ (𝑆𝑣𝑣𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
2120expr 644 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) ∧ (𝑆𝑁)) → ((𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2221an32s 881 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐽) ∧ (𝑆𝑁)) ∧ 𝑣𝐽) → ((𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2322rexlimdva 3169 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐽) ∧ (𝑆𝑁)) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2423ex 449 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐽) → ((𝑆𝑁) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
2524rexlimdva 3169 . . . . 5 (𝐽 ∈ Top → (∃𝐽 (𝑆𝑁) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
2625imp32 448 . . . 4 ((𝐽 ∈ Top ∧ (∃𝐽 (𝑆𝑁) ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
278, 26syldan 488 . . 3 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
28273impb 1108 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
291neiss2 21107 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 𝐽)
301isnei 21109 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
3129, 30syldan 488 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
32313adant3 1127 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
335, 28, 32mpbir2and 995 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 383  w3a 1072   = wceq 1632  wcel 2139  wrex 3051  cin 3714  wss 3715   cuni 4588  cfv 6049  Topctop 20900  neicnei 21103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20901  df-nei 21104
This theorem is referenced by:  neifil  21885  neificl  33862
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