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Theorem sallnei 25529
Description: Two ways to state the set of all the neighborhoods. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
sallnei.1  |-  X  = 
U. J
Assertion
Ref Expression
sallnei  |-  ( J  e.  Top  ->  U. ran  ( nei `  J )  =  { v  |  ( v  C_  X  /\  E. g  e.  J  g  C_  v ) } )
Distinct variable groups:    g, J, v    g, X, v

Proof of Theorem sallnei
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 sallnei.1 . . . 4  |-  X  = 
U. J
21neif 16837 . . 3  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
3 fniunfv 5773 . . . 4  |-  ( ( nei `  J )  Fn  ~P X  ->  U_ s  e.  ~P  X ( ( nei `  J ) `  s
)  =  U. ran  ( nei `  J ) )
43eqcomd 2288 . . 3  |-  ( ( nei `  J )  Fn  ~P X  ->  U. ran  ( nei `  J
)  =  U_ s  e.  ~P  X ( ( nei `  J ) `
 s ) )
52, 4syl 15 . 2  |-  ( J  e.  Top  ->  U. ran  ( nei `  J )  =  U_ s  e. 
~P  X ( ( nei `  J ) `
 s ) )
6 vex 2791 . . . . . 6  |-  s  e. 
_V
76elpw 3631 . . . . 5  |-  ( s  e.  ~P X  <->  s  C_  X )
81neival 16839 . . . . . . 7  |-  ( ( J  e.  Top  /\  s  C_  X )  -> 
( ( nei `  J
) `  s )  =  { v  e.  ~P X  |  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) } )
9 df-rab 2552 . . . . . . . 8  |-  { v  e.  ~P X  |  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) }  =  { v  |  ( v  e. 
~P X  /\  E. g  e.  J  (
s  C_  g  /\  g  C_  v ) ) }
10 vex 2791 . . . . . . . . . . 11  |-  v  e. 
_V
1110elpw 3631 . . . . . . . . . 10  |-  ( v  e.  ~P X  <->  v  C_  X )
1211anbi1i 676 . . . . . . . . 9  |-  ( ( v  e.  ~P X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) )  <->  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v
) ) )
1312abbii 2395 . . . . . . . 8  |-  { v  |  ( v  e. 
~P X  /\  E. g  e.  J  (
s  C_  g  /\  g  C_  v ) ) }  =  { v  |  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v
) ) }
149, 13eqtri 2303 . . . . . . 7  |-  { v  e.  ~P X  |  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) }  =  { v  |  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v
) ) }
158, 14syl6eq 2331 . . . . . 6  |-  ( ( J  e.  Top  /\  s  C_  X )  -> 
( ( nei `  J
) `  s )  =  { v  |  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) ) } )
1615ex 423 . . . . 5  |-  ( J  e.  Top  ->  (
s  C_  X  ->  ( ( nei `  J
) `  s )  =  { v  |  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) ) } ) )
177, 16syl5bi 208 . . . 4  |-  ( J  e.  Top  ->  (
s  e.  ~P X  ->  ( ( nei `  J
) `  s )  =  { v  |  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) ) } ) )
1817ralrimiv 2625 . . 3  |-  ( J  e.  Top  ->  A. s  e.  ~P  X ( ( nei `  J ) `
 s )  =  { v  |  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) ) } )
19 iuneq2 3921 . . 3  |-  ( A. s  e.  ~P  X
( ( nei `  J
) `  s )  =  { v  |  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) ) }  ->  U_ s  e. 
~P  X ( ( nei `  J ) `
 s )  = 
U_ s  e.  ~P  X { v  |  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) ) } )
2018, 19syl 15 . 2  |-  ( J  e.  Top  ->  U_ s  e.  ~P  X ( ( nei `  J ) `
 s )  = 
U_ s  e.  ~P  X { v  |  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) ) } )
21 iunab 3948 . . 3  |-  U_ s  e.  ~P  X { v  |  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v
) ) }  =  { v  |  E. s  e.  ~P  X
( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) ) }
22 simpr 447 . . . . . . . . 9  |-  ( ( s  C_  g  /\  g  C_  v )  -> 
g  C_  v )
2322reximi 2650 . . . . . . . 8  |-  ( E. g  e.  J  ( s  C_  g  /\  g  C_  v )  ->  E. g  e.  J  g  C_  v )
2423anim2i 552 . . . . . . 7  |-  ( ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) )  ->  ( v  C_  X  /\  E. g  e.  J  g  C_  v
) )
2524rexlimivw 2663 . . . . . 6  |-  ( E. s  e.  ~P  X
( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) )  ->  ( v  C_  X  /\  E. g  e.  J  g  C_  v ) )
26 0ss 3483 . . . . . . . . 9  |-  (/)  C_  g
2726biantrur 492 . . . . . . . 8  |-  ( g 
C_  v  <->  ( (/)  C_  g  /\  g  C_  v ) )
2827rexbii 2568 . . . . . . 7  |-  ( E. g  e.  J  g 
C_  v  <->  E. g  e.  J  ( (/)  C_  g  /\  g  C_  v ) )
29 0elpw 4180 . . . . . . . 8  |-  (/)  e.  ~P X
30 sseq1 3199 . . . . . . . . . . . 12  |-  ( s  =  (/)  ->  ( s 
C_  g  <->  (/)  C_  g
) )
3130anbi1d 685 . . . . . . . . . . 11  |-  ( s  =  (/)  ->  ( ( s  C_  g  /\  g  C_  v )  <->  ( (/)  C_  g  /\  g  C_  v ) ) )
3231rexbidv 2564 . . . . . . . . . 10  |-  ( s  =  (/)  ->  ( E. g  e.  J  ( s  C_  g  /\  g  C_  v )  <->  E. g  e.  J  ( (/)  C_  g  /\  g  C_  v ) ) )
3332anbi2d 684 . . . . . . . . 9  |-  ( s  =  (/)  ->  ( ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) )  <-> 
( v  C_  X  /\  E. g  e.  J  ( (/)  C_  g  /\  g  C_  v ) ) ) )
3433rspcev 2884 . . . . . . . 8  |-  ( (
(/)  e.  ~P X  /\  ( v  C_  X  /\  E. g  e.  J  ( (/)  C_  g  /\  g  C_  v ) ) )  ->  E. s  e.  ~P  X ( v 
C_  X  /\  E. g  e.  J  (
s  C_  g  /\  g  C_  v ) ) )
3529, 34mpan 651 . . . . . . 7  |-  ( ( v  C_  X  /\  E. g  e.  J  (
(/)  C_  g  /\  g  C_  v ) )  ->  E. s  e.  ~P  X ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v
) ) )
3628, 35sylan2b 461 . . . . . 6  |-  ( ( v  C_  X  /\  E. g  e.  J  g 
C_  v )  ->  E. s  e.  ~P  X ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v
) ) )
3725, 36impbii 180 . . . . 5  |-  ( E. s  e.  ~P  X
( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v ) )  <->  ( v  C_  X  /\  E. g  e.  J  g  C_  v
) )
3837a1i 10 . . . 4  |-  ( J  e.  Top  ->  ( E. s  e.  ~P  X ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v
) )  <->  ( v  C_  X  /\  E. g  e.  J  g  C_  v ) ) )
3938abbidv 2397 . . 3  |-  ( J  e.  Top  ->  { v  |  E. s  e. 
~P  X ( v 
C_  X  /\  E. g  e.  J  (
s  C_  g  /\  g  C_  v ) ) }  =  { v  |  ( v  C_  X  /\  E. g  e.  J  g  C_  v
) } )
4021, 39syl5eq 2327 . 2  |-  ( J  e.  Top  ->  U_ s  e.  ~P  X { v  |  ( v  C_  X  /\  E. g  e.  J  ( s  C_  g  /\  g  C_  v
) ) }  =  { v  |  ( v  C_  X  /\  E. g  e.  J  g 
C_  v ) } )
415, 20, 403eqtrd 2319 1  |-  ( J  e.  Top  ->  U. ran  ( nei `  J )  =  { v  |  ( v  C_  X  /\  E. g  e.  J  g  C_  v ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   U_ciun 3905   ran crn 4690    Fn wfn 5250   ` cfv 5255   Topctop 16631   neicnei 16834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-nei 16835
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