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Theorem innei 12175
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 2115 . . . . 5 𝐽 = 𝐽
21neii1 12159 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
3 ssinss1 3271 . . . 4 (𝑁 𝐽 → (𝑁𝑀) ⊆ 𝐽)
42, 3syl 14 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ⊆ 𝐽)
543adant3 984 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ⊆ 𝐽)
6 neii2 12161 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝐽 (𝑆𝑁))
7 neii2 12161 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑣𝐽 (𝑆𝑣𝑣𝑀))
86, 7anim12dan 572 . . . 4 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → (∃𝐽 (𝑆𝑁) ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑀)))
9 inopn 12013 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐽𝑣𝐽) → (𝑣) ∈ 𝐽)
1093expa 1164 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) → (𝑣) ∈ 𝐽)
11 ssin 3264 . . . . . . . . . . . . 13 ((𝑆𝑆𝑣) ↔ 𝑆 ⊆ (𝑣))
1211biimpi 119 . . . . . . . . . . . 12 ((𝑆𝑆𝑣) → 𝑆 ⊆ (𝑣))
13 ss2in 3270 . . . . . . . . . . . 12 ((𝑁𝑣𝑀) → (𝑣) ⊆ (𝑁𝑀))
1412, 13anim12i 334 . . . . . . . . . . 11 (((𝑆𝑆𝑣) ∧ (𝑁𝑣𝑀)) → (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀)))
1514an4s 560 . . . . . . . . . 10 (((𝑆𝑁) ∧ (𝑆𝑣𝑣𝑀)) → (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀)))
16 sseq2 3087 . . . . . . . . . . . 12 (𝑔 = (𝑣) → (𝑆𝑔𝑆 ⊆ (𝑣)))
17 sseq1 3086 . . . . . . . . . . . 12 (𝑔 = (𝑣) → (𝑔 ⊆ (𝑁𝑀) ↔ (𝑣) ⊆ (𝑁𝑀)))
1816, 17anbi12d 462 . . . . . . . . . . 11 (𝑔 = (𝑣) → ((𝑆𝑔𝑔 ⊆ (𝑁𝑀)) ↔ (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀))))
1918rspcev 2760 . . . . . . . . . 10 (((𝑣) ∈ 𝐽 ∧ (𝑆 ⊆ (𝑣) ∧ (𝑣) ⊆ (𝑁𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
2010, 15, 19syl2an 285 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) ∧ ((𝑆𝑁) ∧ (𝑆𝑣𝑣𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
2120expr 370 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐽) ∧ 𝑣𝐽) ∧ (𝑆𝑁)) → ((𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2221an32s 540 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐽) ∧ (𝑆𝑁)) ∧ 𝑣𝐽) → ((𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2322rexlimdva 2523 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐽) ∧ (𝑆𝑁)) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀))))
2423rexlimdva2 2526 . . . . 5 (𝐽 ∈ Top → (∃𝐽 (𝑆𝑁) → (∃𝑣𝐽 (𝑆𝑣𝑣𝑀) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
2524imp32 255 . . . 4 ((𝐽 ∈ Top ∧ (∃𝐽 (𝑆𝑁) ∧ ∃𝑣𝐽 (𝑆𝑣𝑣𝑀))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
268, 25syldan 278 . . 3 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
27263impb 1160 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))
281neiss2 12154 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 𝐽)
291isnei 12156 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
3028, 29syldan 278 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
31303adant3 984 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁𝑀) ⊆ 𝐽 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔 ⊆ (𝑁𝑀)))))
325, 27, 31mpbir2and 911 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 945   = wceq 1314  wcel 1463  wrex 2391  cin 3036  wss 3037   cuni 3702  cfv 5081  Topctop 12007  neicnei 12150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-top 12008  df-nei 12151
This theorem is referenced by: (None)
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