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Theorem sallnei 25632
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 16853 . . 3  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
3 fniunfv 5789 . . . 4  |-  ( ( nei `  J )  Fn  ~P X  ->  U_ s  e.  ~P  X ( ( nei `  J ) `  s
)  =  U. ran  ( nei `  J ) )
43eqcomd 2301 . . 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 2804 . . . . . 6  |-  s  e. 
_V
76elpw 3644 . . . . 5  |-  ( s  e.  ~P X  <->  s  C_  X )
81neival 16855 . . . . . . 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 2565 . . . . . . . 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 2804 . . . . . . . . . . 11  |-  v  e. 
_V
1110elpw 3644 . . . . . . . . . 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 2408 . . . . . . . 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 2316 . . . . . . 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 2344 . . . . . 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 2638 . . 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 3937 . . 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 3964 . . 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 2663 . . . . . . . 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 2676 . . . . . 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 3496 . . . . . . . . 9  |-  (/)  C_  g
2726biantrur 492 . . . . . . . 8  |-  ( g 
C_  v  <->  ( (/)  C_  g  /\  g  C_  v ) )
2827rexbii 2581 . . . . . . 7  |-  ( E. g  e.  J  g 
C_  v  <->  E. g  e.  J  ( (/)  C_  g  /\  g  C_  v ) )
29 0elpw 4196 . . . . . . . 8  |-  (/)  e.  ~P X
30 sseq1 3212 . . . . . . . . . . . 12  |-  ( s  =  (/)  ->  ( s 
C_  g  <->  (/)  C_  g
) )
3130anbi1d 685 . . . . . . . . . . 11  |-  ( s  =  (/)  ->  ( ( s  C_  g  /\  g  C_  v )  <->  ( (/)  C_  g  /\  g  C_  v ) ) )
3231rexbidv 2577 . . . . . . . . . 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 2897 . . . . . . . 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 2410 . . 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 2340 . 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 2332 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 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   U_ciun 3921   ran crn 4706    Fn wfn 5266   ` cfv 5271   Topctop 16647   neicnei 16850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-nei 16851
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