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Theorem ssnei2 23033
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 772 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ 𝑀 βŠ† 𝑋)
2 neii2 23025 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
3 sstr2 3987 . . . . . . 7 (𝑔 βŠ† 𝑁 β†’ (𝑁 βŠ† 𝑀 β†’ 𝑔 βŠ† 𝑀))
43com12 32 . . . . . 6 (𝑁 βŠ† 𝑀 β†’ (𝑔 βŠ† 𝑁 β†’ 𝑔 βŠ† 𝑀))
54anim2d 611 . . . . 5 (𝑁 βŠ† 𝑀 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀)))
65reximdv 3167 . . . 4 (𝑁 βŠ† 𝑀 β†’ (βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀)))
72, 6mpan9 506 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ 𝑁 βŠ† 𝑀) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))
87adantrr 716 . 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 23018 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
119isnei 23020 . . . 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 590 . . 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 480 . 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 712 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 setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3067   βŠ† wss 3947  βˆͺ cuni 4908  β€˜cfv 6548  Topctop 22808  neicnei 23014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-top 22809  df-nei 23015
This theorem is referenced by:  topssnei  23041  nllyrest  23403  nllyidm  23406  hausllycmp  23411  cldllycmp  23412  txnlly  23554  neifil  23797  utop2nei  24168  cnllycmp  24895  gneispb  43561
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