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Theorem neips 22516
Description: A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.)
Hypothesis
Ref Expression
neips.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
neips ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝})))
Distinct variable groups:   𝐽,𝑝   𝑁,𝑝   𝑆,𝑝   𝑋,𝑝
Proof of Theorem neips
Dummy variables 𝑔 β„Ž 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snssi 4788 . . . . . 6 (𝑝 ∈ 𝑆 β†’ {𝑝} βŠ† 𝑆)
2 neiss 22512 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ {𝑝} βŠ† 𝑆) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}))
31, 2syl3an3 1165 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑝 ∈ 𝑆) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}))
433exp 1119 . . . 4 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ (𝑝 ∈ 𝑆 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}))))
54ralrimdv 3151 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝})))
653ad2ant1 1133 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝})))
7 r19.28zv 4478 . . . . 5 (𝑆 β‰  βˆ… β†’ (βˆ€π‘ ∈ 𝑆 (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) ↔ (𝑁 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
873ad2ant3 1135 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (βˆ€π‘ ∈ 𝑆 (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) ↔ (𝑁 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
9 ssrab2 4057 . . . . . . . . . 10 {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝐽
10 uniopn 22298 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝐽) β†’ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∈ 𝐽)
119, 10mpan2 689 . . . . . . . . 9 (𝐽 ∈ Top β†’ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∈ 𝐽)
1211ad2antrr 724 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∈ 𝐽)
13 sseq1 3987 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑔 β†’ (𝑣 βŠ† 𝑁 ↔ 𝑔 βŠ† 𝑁))
1413elrab 3663 . . . . . . . . . . . . . . 15 (𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ↔ (𝑔 ∈ 𝐽 ∧ 𝑔 βŠ† 𝑁))
15 elunii 4890 . . . . . . . . . . . . . . 15 ((𝑝 ∈ 𝑔 ∧ 𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁}) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
1614, 15sylan2br 595 . . . . . . . . . . . . . 14 ((𝑝 ∈ 𝑔 ∧ (𝑔 ∈ 𝐽 ∧ 𝑔 βŠ† 𝑁)) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
1716an12s 647 . . . . . . . . . . . . 13 ((𝑔 ∈ 𝐽 ∧ (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
1817rexlimiva 3146 . . . . . . . . . . . 12 (βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
1918ralimi 3082 . . . . . . . . . . 11 (βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆ€π‘ ∈ 𝑆 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
20 dfss3 3950 . . . . . . . . . . 11 (𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ↔ βˆ€π‘ ∈ 𝑆 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
2119, 20sylibr 233 . . . . . . . . . 10 (βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ 𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
2221adantl 482 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ 𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
23 unissb 4920 . . . . . . . . . 10 (βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁 ↔ βˆ€β„Ž ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁}β„Ž βŠ† 𝑁)
24 sseq1 3987 . . . . . . . . . . . 12 (𝑣 = β„Ž β†’ (𝑣 βŠ† 𝑁 ↔ β„Ž βŠ† 𝑁))
2524elrab 3663 . . . . . . . . . . 11 (β„Ž ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ↔ (β„Ž ∈ 𝐽 ∧ β„Ž βŠ† 𝑁))
2625simprbi 497 . . . . . . . . . 10 (β„Ž ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} β†’ β„Ž βŠ† 𝑁)
2723, 26mprgbir 3067 . . . . . . . . 9 βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁
2822, 27jctir 521 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∧ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁))
29 sseq2 3988 . . . . . . . . . 10 (β„Ž = βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} β†’ (𝑆 βŠ† β„Ž ↔ 𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁}))
30 sseq1 3987 . . . . . . . . . 10 (β„Ž = βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} β†’ (β„Ž βŠ† 𝑁 ↔ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁))
3129, 30anbi12d 631 . . . . . . . . 9 (β„Ž = βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} β†’ ((𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁) ↔ (𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∧ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁)))
3231rspcev 3595 . . . . . . . 8 ((βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∈ 𝐽 ∧ (𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∧ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁)) β†’ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))
3312, 28, 32syl2anc 584 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))
3433ex 413 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁)))
3534anim2d 612 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((𝑁 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑁 βŠ† 𝑋 ∧ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))))
36353adant3 1132 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ ((𝑁 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑁 βŠ† 𝑋 ∧ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))))
378, 36sylbid 239 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (βˆ€π‘ ∈ 𝑆 (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑁 βŠ† 𝑋 ∧ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))))
38 ssel2 3957 . . . . . . 7 ((𝑆 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑆) β†’ 𝑝 ∈ 𝑋)
39 neips.1 . . . . . . . 8 𝑋 = βˆͺ 𝐽
4039isneip 22508 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
4138, 40sylan2 593 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑆)) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
4241anassrs 468 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ 𝑝 ∈ 𝑆) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
4342ralbidva 3174 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) ↔ βˆ€π‘ ∈ 𝑆 (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
44433adant3 1132 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) ↔ βˆ€π‘ ∈ 𝑆 (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
4539isnei 22506 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))))
46453adant3 1132 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))))
4737, 44, 463imtr4d 293 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
486, 47impbid 211 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069  {crab 3418   βŠ† wss 3928  βˆ…c0 4302  {csn 4606  βˆͺ cuni 4885  β€˜cfv 6516  Topctop 22294  neicnei 22500
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 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-top 22295  df-nei 22501
This theorem is referenced by:  utop2nei  23654
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