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Theorem innei 22620
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 2732 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
21neii1 22601 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑁 βŠ† βˆͺ 𝐽)
3 ssinss1 4236 . . . 4 (𝑁 βŠ† βˆͺ 𝐽 β†’ (𝑁 ∩ 𝑀) βŠ† βˆͺ 𝐽)
42, 3syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (𝑁 ∩ 𝑀) βŠ† βˆͺ 𝐽)
543adant3 1132 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (𝑁 ∩ 𝑀) βŠ† βˆͺ 𝐽)
6 neii2 22603 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))
7 neii2 22603 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑀))
86, 7anim12dan 619 . . . 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 22392 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ β„Ž ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) β†’ (β„Ž ∩ 𝑣) ∈ 𝐽)
1093expa 1118 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ β„Ž ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) β†’ (β„Ž ∩ 𝑣) ∈ 𝐽)
11 ssin 4229 . . . . . . . . . . . . 13 ((𝑆 βŠ† β„Ž ∧ 𝑆 βŠ† 𝑣) ↔ 𝑆 βŠ† (β„Ž ∩ 𝑣))
1211biimpi 215 . . . . . . . . . . . 12 ((𝑆 βŠ† β„Ž ∧ 𝑆 βŠ† 𝑣) β†’ 𝑆 βŠ† (β„Ž ∩ 𝑣))
13 ss2in 4235 . . . . . . . . . . . 12 ((β„Ž βŠ† 𝑁 ∧ 𝑣 βŠ† 𝑀) β†’ (β„Ž ∩ 𝑣) βŠ† (𝑁 ∩ 𝑀))
1412, 13anim12i 613 . . . . . . . . . . 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 658 . . . . . . . . . 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 4007 . . . . . . . . . . . 12 (𝑔 = (β„Ž ∩ 𝑣) β†’ (𝑆 βŠ† 𝑔 ↔ 𝑆 βŠ† (β„Ž ∩ 𝑣)))
17 sseq1 4006 . . . . . . . . . . . 12 (𝑔 = (β„Ž ∩ 𝑣) β†’ (𝑔 βŠ† (𝑁 ∩ 𝑀) ↔ (β„Ž ∩ 𝑣) βŠ† (𝑁 ∩ 𝑀)))
1816, 17anbi12d 631 . . . . . . . . . . 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 3612 . . . . . . . . . 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 596 . . . . . . . . 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 457 . . . . . . . 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 650 . . . . . . 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 3155 . . . . . 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 3157 . . . . 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 419 . . . 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 591 . . 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 1115 . 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 22596 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† βˆͺ 𝐽)
291isnei 22598 . . . 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 591 . . 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 1132 . 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 711 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 setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   ∩ cin 3946   βŠ† wss 3947  βˆͺ cuni 4907  β€˜cfv 6540  Topctop 22386  neicnei 22592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 22387  df-nei 22593
This theorem is referenced by:  neifil  23375  neificl  36609
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