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Theorem innei 13666
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 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (𝑁 ∩ 𝑀) ∈ ((neiβ€˜π½)β€˜π‘†))
Proof of Theorem innei
Dummy variables 𝑔 β„Ž 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
21neii1 13650 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑁 βŠ† βˆͺ 𝐽)
3 ssinss1 3365 . . . 4 (𝑁 βŠ† βˆͺ 𝐽 β†’ (𝑁 ∩ 𝑀) βŠ† βˆͺ 𝐽)
42, 3syl 14 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (𝑁 ∩ 𝑀) βŠ† βˆͺ 𝐽)
543adant3 1017 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (𝑁 ∩ 𝑀) βŠ† βˆͺ 𝐽)
6 neii2 13652 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))
7 neii2 13652 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑀))
86, 7anim12dan 600 . . . 4 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ (βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁) ∧ βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑀)))
9 inopn 13506 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ β„Ž ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) β†’ (β„Ž ∩ 𝑣) ∈ 𝐽)
1093expa 1203 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ β„Ž ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) β†’ (β„Ž ∩ 𝑣) ∈ 𝐽)
11 ssin 3358 . . . . . . . . . . . . 13 ((𝑆 βŠ† β„Ž ∧ 𝑆 βŠ† 𝑣) ↔ 𝑆 βŠ† (β„Ž ∩ 𝑣))
1211biimpi 120 . . . . . . . . . . . 12 ((𝑆 βŠ† β„Ž ∧ 𝑆 βŠ† 𝑣) β†’ 𝑆 βŠ† (β„Ž ∩ 𝑣))
13 ss2in 3364 . . . . . . . . . . . 12 ((β„Ž βŠ† 𝑁 ∧ 𝑣 βŠ† 𝑀) β†’ (β„Ž ∩ 𝑣) βŠ† (𝑁 ∩ 𝑀))
1412, 13anim12i 338 . . . . . . . . . . 11 (((𝑆 βŠ† β„Ž ∧ 𝑆 βŠ† 𝑣) ∧ (β„Ž βŠ† 𝑁 ∧ 𝑣 βŠ† 𝑀)) β†’ (𝑆 βŠ† (β„Ž ∩ 𝑣) ∧ (β„Ž ∩ 𝑣) βŠ† (𝑁 ∩ 𝑀)))
1514an4s 588 . . . . . . . . . 10 (((𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁) ∧ (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑀)) β†’ (𝑆 βŠ† (β„Ž ∩ 𝑣) ∧ (β„Ž ∩ 𝑣) βŠ† (𝑁 ∩ 𝑀)))
16 sseq2 3180 . . . . . . . . . . . 12 (𝑔 = (β„Ž ∩ 𝑣) β†’ (𝑆 βŠ† 𝑔 ↔ 𝑆 βŠ† (β„Ž ∩ 𝑣)))
17 sseq1 3179 . . . . . . . . . . . 12 (𝑔 = (β„Ž ∩ 𝑣) β†’ (𝑔 βŠ† (𝑁 ∩ 𝑀) ↔ (β„Ž ∩ 𝑣) βŠ† (𝑁 ∩ 𝑀)))
1816, 17anbi12d 473 . . . . . . . . . . 11 (𝑔 = (β„Ž ∩ 𝑣) β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀)) ↔ (𝑆 βŠ† (β„Ž ∩ 𝑣) ∧ (β„Ž ∩ 𝑣) βŠ† (𝑁 ∩ 𝑀))))
1918rspcev 2842 . . . . . . . . . 10 (((β„Ž ∩ 𝑣) ∈ 𝐽 ∧ (𝑆 βŠ† (β„Ž ∩ 𝑣) ∧ (β„Ž ∩ 𝑣) βŠ† (𝑁 ∩ 𝑀))) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀)))
2010, 15, 19syl2an 289 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ β„Ž ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ ((𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁) ∧ (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑀))) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀)))
2120expr 375 . . . . . . . 8 ((((𝐽 ∈ Top ∧ β„Ž ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁)) β†’ ((𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑀) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀))))
2221an32s 568 . . . . . . 7 ((((𝐽 ∈ Top ∧ β„Ž ∈ 𝐽) ∧ (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁)) ∧ 𝑣 ∈ 𝐽) β†’ ((𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑀) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀))))
2322rexlimdva 2594 . . . . . 6 (((𝐽 ∈ Top ∧ β„Ž ∈ 𝐽) ∧ (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁)) β†’ (βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑀) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀))))
2423rexlimdva2 2597 . . . . 5 (𝐽 ∈ Top β†’ (βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁) β†’ (βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑀) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀)))))
2524imp32 257 . . . 4 ((𝐽 ∈ Top ∧ (βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁) ∧ βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑀))) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀)))
268, 25syldan 282 . . 3 ((𝐽 ∈ Top ∧ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†))) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀)))
27263impb 1199 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀)))
281neiss2 13645 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† βˆͺ 𝐽)
291isnei 13647 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((𝑁 ∩ 𝑀) ∈ ((neiβ€˜π½)β€˜π‘†) ↔ ((𝑁 ∩ 𝑀) βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀)))))
3028, 29syldan 282 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ ((𝑁 ∩ 𝑀) ∈ ((neiβ€˜π½)β€˜π‘†) ↔ ((𝑁 ∩ 𝑀) βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀)))))
31303adant3 1017 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ ((𝑁 ∩ 𝑀) ∈ ((neiβ€˜π½)β€˜π‘†) ↔ ((𝑁 ∩ 𝑀) βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† (𝑁 ∩ 𝑀)))))
325, 27, 31mpbir2and 944 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (𝑁 ∩ 𝑀) ∈ ((neiβ€˜π½)β€˜π‘†))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆƒwrex 2456   ∩ cin 3129   βŠ† wss 3130  βˆͺ cuni 3810  β€˜cfv 5217  Topctop 13500  neicnei 13641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-top 13501  df-nei 13642
This theorem is referenced by: (None)
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