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Theorem ssnei2 22611
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 771 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ (𝑁 βŠ† 𝑀 ∧ 𝑀 βŠ† 𝑋)) β†’ 𝑀 βŠ† 𝑋)
2 neii2 22603 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))
3 sstr2 3988 . . . . . . 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 612 . . . . 5 (𝑁 βŠ† 𝑀 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀)))
65reximdv 3170 . . . 4 (𝑁 βŠ† 𝑀 β†’ (βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀)))
72, 6mpan9 507 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) ∧ 𝑁 βŠ† 𝑀) β†’ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑀))
87adantrr 715 . 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 22596 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
119isnei 22598 . . . 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 591 . . 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 481 . 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 711 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 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βŠ† wss 3947  βˆͺ cuni 4907  β€˜cfv 6540  Topctop 22386  neicnei 22592
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 22387  df-nei 22593
This theorem is referenced by:  topssnei  22619  nllyrest  22981  nllyidm  22984  hausllycmp  22989  cldllycmp  22990  txnlly  23132  neifil  23375  utop2nei  23746  cnllycmp  24463  gneispb  42867
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