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Theorem neiss 13735
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 2177 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
21neii1 13732 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑁 βŠ† βˆͺ 𝐽)
323adant3 1017 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝑁 βŠ† βˆͺ 𝐽)
4 neii2 13734 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
543adant3 1017 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
6 sstr2 3164 . . . . . 6 (𝑅 βŠ† 𝑆 β†’ (𝑆 βŠ† 𝑔 β†’ 𝑅 βŠ† 𝑔))
76anim1d 336 . . . . 5 (𝑅 βŠ† 𝑆 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ (𝑅 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
87reximdv 2578 . . . 4 (𝑅 βŠ† 𝑆 β†’ (βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑅 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
983ad2ant3 1020 . . 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 13 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ βˆƒπ‘” ∈ 𝐽 (𝑅 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
11 simp1 997 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝐽 ∈ Top)
12 simp3 999 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝑅 βŠ† 𝑆)
131neiss2 13727 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† βˆͺ 𝐽)
14133adant3 1017 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝑆 βŠ† βˆͺ 𝐽)
1512, 14sstrd 3167 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝑅 βŠ† βˆͺ 𝐽)
161isnei 13729 . . 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 411 . 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 944 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑅 βŠ† 𝑆) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘…))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   ∈ wcel 2148  βˆƒwrex 2456   βŠ† wss 3131  βˆͺ cuni 3811  β€˜cfv 5218  Topctop 13582  neicnei 13723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-top 13583  df-nei 13724
This theorem is referenced by:  neipsm  13739  neissex  13750
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