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Theorem neips 22839
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 4812 . . . . . 6 (𝑝 ∈ 𝑆 β†’ {𝑝} βŠ† 𝑆)
2 neiss 22835 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ {𝑝} βŠ† 𝑆) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}))
31, 2syl3an3 1163 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑝 ∈ 𝑆) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}))
433exp 1117 . . . 4 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ (𝑝 ∈ 𝑆 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}))))
54ralrimdv 3150 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝})))
653ad2ant1 1131 . 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 4501 . . . . 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 1133 . . . 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 4078 . . . . . . . . . 10 {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝐽
10 uniopn 22621 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝐽) β†’ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∈ 𝐽)
119, 10mpan2 687 . . . . . . . . 9 (𝐽 ∈ Top β†’ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∈ 𝐽)
1211ad2antrr 722 . . . . . . . 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 4008 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑔 β†’ (𝑣 βŠ† 𝑁 ↔ 𝑔 βŠ† 𝑁))
1413elrab 3684 . . . . . . . . . . . . . . 15 (𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ↔ (𝑔 ∈ 𝐽 ∧ 𝑔 βŠ† 𝑁))
15 elunii 4914 . . . . . . . . . . . . . . 15 ((𝑝 ∈ 𝑔 ∧ 𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁}) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
1614, 15sylan2br 593 . . . . . . . . . . . . . 14 ((𝑝 ∈ 𝑔 ∧ (𝑔 ∈ 𝐽 ∧ 𝑔 βŠ† 𝑁)) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
1716an12s 645 . . . . . . . . . . . . 13 ((𝑔 ∈ 𝐽 ∧ (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
1817rexlimiva 3145 . . . . . . . . . . . 12 (βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
1918ralimi 3081 . . . . . . . . . . 11 (βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆ€π‘ ∈ 𝑆 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
20 dfss3 3971 . . . . . . . . . . 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 480 . . . . . . . . 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 4944 . . . . . . . . . 10 (βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁 ↔ βˆ€β„Ž ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁}β„Ž βŠ† 𝑁)
24 sseq1 4008 . . . . . . . . . . . 12 (𝑣 = β„Ž β†’ (𝑣 βŠ† 𝑁 ↔ β„Ž βŠ† 𝑁))
2524elrab 3684 . . . . . . . . . . 11 (β„Ž ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ↔ (β„Ž ∈ 𝐽 ∧ β„Ž βŠ† 𝑁))
2625simprbi 495 . . . . . . . . . 10 (β„Ž ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} β†’ β„Ž βŠ† 𝑁)
2723, 26mprgbir 3066 . . . . . . . . 9 βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁
2822, 27jctir 519 . . . . . . . 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 4009 . . . . . . . . . 10 (β„Ž = βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} β†’ (𝑆 βŠ† β„Ž ↔ 𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁}))
30 sseq1 4008 . . . . . . . . . 10 (β„Ž = βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} β†’ (β„Ž βŠ† 𝑁 ↔ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁))
3129, 30anbi12d 629 . . . . . . . . 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 3613 . . . . . . . 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 582 . . . . . . 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 411 . . . . . 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 610 . . . . 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 1130 . . . 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 3978 . . . . . . 7 ((𝑆 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑆) β†’ 𝑝 ∈ 𝑋)
39 neips.1 . . . . . . . 8 𝑋 = βˆͺ 𝐽
4039isneip 22831 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
4138, 40sylan2 591 . . . . . 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 466 . . . . 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 3173 . . . 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 1130 . . 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 22829 . . . 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 1130 . . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22617  neicnei 22823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22618  df-nei 22824
This theorem is referenced by:  utop2nei  23977
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