MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnei Structured version   Visualization version   GIF version
Theorem isnei 22598
Description: The predicate "the class 𝑁 is a neighborhood of 𝑆". (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
neifval.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
isnei ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
Distinct variable groups:   𝑔,𝐽   𝑔,𝑁   𝑆,𝑔   𝑔,𝑋
Proof of Theorem isnei
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . 4 𝑋 = βˆͺ 𝐽
21neival 22597 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((neiβ€˜π½)β€˜π‘†) = {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)})
32eleq2d 2819 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ 𝑁 ∈ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)}))
4 sseq2 4007 . . . . . . 7 (𝑣 = 𝑁 β†’ (𝑔 βŠ† 𝑣 ↔ 𝑔 βŠ† 𝑁))
54anbi2d 629 . . . . . 6 (𝑣 = 𝑁 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣) ↔ (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
65rexbidv 3178 . . . . 5 (𝑣 = 𝑁 β†’ (βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣) ↔ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
76elrab 3682 . . . 4 (𝑁 ∈ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ↔ (𝑁 ∈ 𝒫 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
81topopn 22399 . . . . . 6 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
9 elpw2g 5343 . . . . . 6 (𝑋 ∈ 𝐽 β†’ (𝑁 ∈ 𝒫 𝑋 ↔ 𝑁 βŠ† 𝑋))
108, 9syl 17 . . . . 5 (𝐽 ∈ Top β†’ (𝑁 ∈ 𝒫 𝑋 ↔ 𝑁 βŠ† 𝑋))
1110anbi1d 630 . . . 4 (𝐽 ∈ Top β†’ ((𝑁 ∈ 𝒫 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
127, 11bitrid 282 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
1312adantr 481 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑁 ∈ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
143, 13bitrd 278 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432   βŠ† wss 3947  π’« cpw 4601  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:  neiint  22599  isneip  22600  neii1  22601  neii2  22603  neiss  22604  neips  22608  opnneissb  22609  opnssneib  22610  ssnei2  22611  innei  22620  neitr  22675  neitx  23102  neifg  35244  islptre  44321  sepfsepc  47513
  Copyright terms: Public domain W3C validator