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Theorem isnei 22607
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 22606 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((neiβ€˜π½)β€˜π‘†) = {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)})
32eleq2d 2820 . 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 4009 . . . . . . 7 (𝑣 = 𝑁 β†’ (𝑔 βŠ† 𝑣 ↔ 𝑔 βŠ† 𝑁))
54anbi2d 630 . . . . . 6 (𝑣 = 𝑁 β†’ ((𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣) ↔ (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
65rexbidv 3179 . . . . 5 (𝑣 = 𝑁 β†’ (βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣) ↔ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑁)))
76elrab 3684 . . . 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 22408 . . . . . 6 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
9 elpw2g 5345 . . . . . 6 (𝑋 ∈ 𝐽 β†’ (𝑁 ∈ 𝒫 𝑋 ↔ 𝑁 βŠ† 𝑋))
108, 9syl 17 . . . . 5 (𝐽 ∈ Top β†’ (𝑁 ∈ 𝒫 𝑋 ↔ 𝑁 βŠ† 𝑋))
1110anbi1d 631 . . . 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 283 . . 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 482 . 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 279 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  {crab 3433   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  neicnei 22601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22396  df-nei 22602
This theorem is referenced by:  neiint  22608  isneip  22609  neii1  22610  neii2  22612  neiss  22613  neips  22617  opnneissb  22618  opnssneib  22619  ssnei2  22620  innei  22629  neitr  22684  neitx  23111  neifg  35256  islptre  44335  sepfsepc  47560
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