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Theorem neiss 22833
Description: Any neighborhood of a set 𝑆 is also a neighborhood of any subset 𝑅 βŠ† 𝑆. Similar to Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.)
Assertion
Ref Expression
neiss ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘…))
Proof of Theorem neiss
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
21neii1 22830 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑁 βŠ† βˆͺ 𝐽)
323adant3 1132 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝑁 βŠ† βˆͺ 𝐽)
4 neii2 22832 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
543adant3 1132 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
6 sstr2 3989 . . . . . 6 (𝑅 βŠ† 𝑆 β†’ (𝑆 βŠ† 𝑔 β†’ 𝑅 βŠ† 𝑔))
76anim1d 611 . . . . 5 (𝑅 βŠ† 𝑆 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ (𝑅 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
87reximdv 3170 . . . 4 (𝑅 βŠ† 𝑆 β†’ (βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑅 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
983ad2ant3 1135 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ (βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑅 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
105, 9mpd 15 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ βˆƒπ‘” ∈ 𝐽 (𝑅 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
11 simp1 1136 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝐽 ∈ Top)
12 simp3 1138 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝑅 βŠ† 𝑆)
131neiss2 22825 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† βˆͺ 𝐽)
14133adant3 1132 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝑆 βŠ† βˆͺ 𝐽)
1512, 14sstrd 3992 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝑅 βŠ† βˆͺ 𝐽)
161isnei 22827 . . 3 ((𝐽 ∈ Top ∧ 𝑅 βŠ† βˆͺ 𝐽) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘…) ↔ (𝑁 βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘” ∈ 𝐽 (𝑅 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
1711, 15, 16syl2anc 584 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘…) ↔ (𝑁 βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘” ∈ 𝐽 (𝑅 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
183, 10, 17mpbir2and 711 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘…))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   ∈ wcel 2106  βˆƒwrex 3070   βŠ† wss 3948  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22615  neicnei 22821
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22616  df-nei 22822
This theorem is referenced by:  neips  22837  neissex  22851
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