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Theorem neissex 23051
Description: For any neighborhood 𝑁 of 𝑆, there is a neighborhood π‘₯ of 𝑆 such that 𝑁 is a neighborhood of all subsets of π‘₯. Generalization to subsets of Property Viv of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
Assertion
Ref Expression
neissex ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)βˆ€π‘¦(𝑦 βŠ† π‘₯ β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘¦)))
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝑁,𝑦   π‘₯,𝑆,𝑦

Proof of Theorem neissex
StepHypRef Expression
1 neii2 23032 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑆 βŠ† π‘₯ ∧ π‘₯ βŠ† 𝑁))
2 opnneiss 23042 . . . . 5 ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝑆 βŠ† π‘₯) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))
323expb 1117 . . . 4 ((𝐽 ∈ Top ∧ (π‘₯ ∈ 𝐽 ∧ 𝑆 βŠ† π‘₯)) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))
43adantrrr 723 . . 3 ((𝐽 ∈ Top ∧ (π‘₯ ∈ 𝐽 ∧ (𝑆 βŠ† π‘₯ ∧ π‘₯ βŠ† 𝑁))) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))
54adantlr 713 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (π‘₯ ∈ 𝐽 ∧ (𝑆 βŠ† π‘₯ ∧ π‘₯ βŠ† 𝑁))) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜π‘†))
6 simplll 773 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (π‘₯ ∈ 𝐽 ∧ π‘₯ βŠ† 𝑁)) ∧ 𝑦 βŠ† π‘₯) β†’ 𝐽 ∈ Top)
7 simpll 765 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝐽) β†’ 𝐽 ∈ Top)
8 simpr 483 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝐽) β†’ π‘₯ ∈ 𝐽)
9 eqid 2728 . . . . . . . . . . . 12 βˆͺ 𝐽 = βˆͺ 𝐽
109neii1 23030 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑁 βŠ† βˆͺ 𝐽)
1110adantr 479 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝐽) β†’ 𝑁 βŠ† βˆͺ 𝐽)
129opnssneib 23039 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝑁 βŠ† βˆͺ 𝐽) β†’ (π‘₯ βŠ† 𝑁 ↔ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘₯)))
137, 8, 11, 12syl3anc 1368 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝐽) β†’ (π‘₯ βŠ† 𝑁 ↔ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘₯)))
1413biimpa 475 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ 𝐽) ∧ π‘₯ βŠ† 𝑁) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘₯))
1514anasss 465 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (π‘₯ ∈ 𝐽 ∧ π‘₯ βŠ† 𝑁)) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘₯))
1615adantr 479 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (π‘₯ ∈ 𝐽 ∧ π‘₯ βŠ† 𝑁)) ∧ 𝑦 βŠ† π‘₯) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘₯))
17 simpr 483 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (π‘₯ ∈ 𝐽 ∧ π‘₯ βŠ† 𝑁)) ∧ 𝑦 βŠ† π‘₯) β†’ 𝑦 βŠ† π‘₯)
18 neiss 23033 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘₯) ∧ 𝑦 βŠ† π‘₯) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘¦))
196, 16, 17, 18syl3anc 1368 . . . . 5 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (π‘₯ ∈ 𝐽 ∧ π‘₯ βŠ† 𝑁)) ∧ 𝑦 βŠ† π‘₯) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘¦))
2019ex 411 . . . 4 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (π‘₯ ∈ 𝐽 ∧ π‘₯ βŠ† 𝑁)) β†’ (𝑦 βŠ† π‘₯ β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘¦)))
2120adantrrl 722 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (π‘₯ ∈ 𝐽 ∧ (𝑆 βŠ† π‘₯ ∧ π‘₯ βŠ† 𝑁))) β†’ (𝑦 βŠ† π‘₯ β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘¦)))
2221alrimiv 1922 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (π‘₯ ∈ 𝐽 ∧ (𝑆 βŠ† π‘₯ ∧ π‘₯ βŠ† 𝑁))) β†’ βˆ€π‘¦(𝑦 βŠ† π‘₯ β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘¦)))
231, 5, 22reximssdv 3170 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘₯ ∈ ((neiβ€˜π½)β€˜π‘†)βˆ€π‘¦(𝑦 βŠ† π‘₯ β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘¦)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394  βˆ€wal 1531   ∈ wcel 2098  βˆƒwrex 3067   βŠ† wss 3949  βˆͺ cuni 4912  β€˜cfv 6553  Topctop 22815  neicnei 23021
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-top 22816  df-nei 23022
This theorem is referenced by: (None)
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