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Theorem innei 22973
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 2724 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
21neii1 22954 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑁 βŠ† βˆͺ 𝐽)
3 ssinss1 4230 . . . 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 1129 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (𝑁 ∩ 𝑀) βŠ† βˆͺ 𝐽)
6 neii2 22956 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))
7 neii2 22956 . . . . 5 ((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘£ ∈ 𝐽 (𝑆 βŠ† 𝑣 ∧ 𝑣 βŠ† 𝑀))
86, 7anim12dan 618 . . . 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 22745 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ β„Ž ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) β†’ (β„Ž ∩ 𝑣) ∈ 𝐽)
1093expa 1115 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ β„Ž ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) β†’ (β„Ž ∩ 𝑣) ∈ 𝐽)
11 ssin 4223 . . . . . . . . . . . . 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 4229 . . . . . . . . . . . 12 ((β„Ž βŠ† 𝑁 ∧ 𝑣 βŠ† 𝑀) β†’ (β„Ž ∩ 𝑣) βŠ† (𝑁 ∩ 𝑀))
1412, 13anim12i 612 . . . . . . . . . . 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 657 . . . . . . . . . 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 4001 . . . . . . . . . . . 12 (𝑔 = (β„Ž ∩ 𝑣) β†’ (𝑆 βŠ† 𝑔 ↔ 𝑆 βŠ† (β„Ž ∩ 𝑣)))
17 sseq1 4000 . . . . . . . . . . . 12 (𝑔 = (β„Ž ∩ 𝑣) β†’ (𝑔 βŠ† (𝑁 ∩ 𝑀) ↔ (β„Ž ∩ 𝑣) βŠ† (𝑁 ∩ 𝑀)))
1816, 17anbi12d 630 . . . . . . . . . . 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 3604 . . . . . . . . . 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 595 . . . . . . . . 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 456 . . . . . . . 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 649 . . . . . . 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 3147 . . . . . 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 3149 . . . . 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 418 . . . 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 590 . . 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 1112 . 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 22949 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† βˆͺ 𝐽)
291isnei 22951 . . . 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 590 . . 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 1129 . 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 710 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3062   ∩ cin 3940   βŠ† wss 3941  βˆͺ cuni 4900  β€˜cfv 6534  Topctop 22739  neicnei 22945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-top 22740  df-nei 22946
This theorem is referenced by:  neifil  23728  neificl  37125
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