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Theorem ssnei2 13660
Description: Any subset 𝑀 of 𝑋 containing a neighborhood 𝑁 of a set 𝑆 is a neighborhood of this set. Generalization to subsets of Property Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
Hypothesis
Ref Expression
neips.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
ssnei2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†))
Proof of Theorem ssnei2
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 simprr 531 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ 𝑀 βŠ† 𝑋)
2 neii2 13652 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
3 sstr2 3163 . . . . . . 7 (𝑔 βŠ† 𝑁 β†’ (𝑁 βŠ† 𝑀 β†’ 𝑔 βŠ† 𝑀))
43com12 30 . . . . . 6 (𝑁 βŠ† 𝑀 β†’ (𝑔 βŠ† 𝑁 β†’ 𝑔 βŠ† 𝑀))
54anim2d 337 . . . . 5 (𝑁 βŠ† 𝑀 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀)))
65reximdv 2578 . . . 4 (𝑁 βŠ† 𝑀 β†’ (βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀)))
72, 6mpan9 281 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ 𝑁 βŠ† 𝑀) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))
87adantrr 479 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))
9 neips.1 . . . . 5 𝑋 = βˆͺ 𝐽
109neiss2 13645 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
119isnei 13647 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑀 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑀 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))))
1210, 11syldan 282 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ (𝑀 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑀 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))))
1312adantr 276 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ (𝑀 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑀 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))))
141, 8, 13mpbir2and 944 1 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ 𝑀 ∈ ((neiβ€˜π½)β€˜π‘†))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  βˆƒwrex 2456   βŠ† wss 3130  βˆͺ cuni 3810  β€˜cfv 5217  Topctop 13500  neicnei 13641
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 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210
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 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-top 13501  df-nei 13642
This theorem is referenced by:  topssnei  13665
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