MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neifval Structured version   Unicode version

Theorem neifval 18703
Description: The neighborhood function on the subsets of a topology's base set. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
neifval.1  |-  X  = 
U. J
Assertion
Ref Expression
neifval  |-  ( J  e.  Top  ->  ( nei `  J )  =  ( x  e.  ~P X  |->  { v  e. 
~P X  |  E. g  e.  J  (
x  C_  g  /\  g  C_  v ) } ) )
Distinct variable groups:    v, g, x, J    g, X, v, x

Proof of Theorem neifval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . 4  |-  X  = 
U. J
21topopn 18519 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4476 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5947 . . 3  |-  ( ~P X  e.  _V  ->  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )  e.  _V )
52, 3, 43syl 20 . 2  |-  ( J  e.  Top  ->  (
x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )  e.  _V )
6 unieq 4099 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2493 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3865 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 rexeq 2918 . . . . 5  |-  ( j  =  J  ->  ( E. g  e.  j 
( x  C_  g  /\  g  C_  v )  <->  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) ) )
108, 9rabeqbidv 2967 . . . 4  |-  ( j  =  J  ->  { v  e.  ~P U. j  |  E. g  e.  j  ( x  C_  g  /\  g  C_  v ) }  =  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )
118, 10mpteq12dv 4370 . . 3  |-  ( j  =  J  ->  (
x  e.  ~P U. j  |->  { v  e. 
~P U. j  |  E. g  e.  j  (
x  C_  g  /\  g  C_  v ) } )  =  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } ) )
12 df-nei 18702 . . 3  |-  nei  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  { v  e. 
~P U. j  |  E. g  e.  j  (
x  C_  g  /\  g  C_  v ) } ) )
1311, 12fvmptg 5772 . 2  |-  ( ( J  e.  Top  /\  ( x  e.  ~P X  |->  { v  e. 
~P X  |  E. g  e.  J  (
x  C_  g  /\  g  C_  v ) } )  e.  _V )  ->  ( nei `  J
)  =  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } ) )
145, 13mpdan 668 1  |-  ( J  e.  Top  ->  ( nei `  J )  =  ( x  e.  ~P X  |->  { v  e. 
~P X  |  E. g  e.  J  (
x  C_  g  /\  g  C_  v ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716   {crab 2719   _Vcvv 2972    C_ wss 3328   ~Pcpw 3860   U.cuni 4091    e. cmpt 4350   ` cfv 5418   Topctop 18498   neicnei 18701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-top 18503  df-nei 18702
This theorem is referenced by:  neif  18704  neival  18706
  Copyright terms: Public domain W3C validator