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

Theorem kgenval 17246
Description: Value of the compact generator. (The "k" in 𝑘Gen comes from the name "k-space" for these spaces, after the German word kompakt.) (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
kgenval  |-  ( J  e.  (TopOn `  X
)  ->  (𝑘Gen `  J
)  =  { x  e.  ~P X  |  A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) } )
Distinct variable groups:    x, k, J    k, X, x

Proof of Theorem kgenval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 df-kgen 17245 . . 3  |- 𝑘Gen  =  (
j  e.  Top  |->  { x  e.  ~P U. j  |  A. k  e.  ~P  U. j ( ( jt  k )  e. 
Comp  ->  ( x  i^i  k )  e.  ( jt  k ) ) } )
21a1i 10 . 2  |-  ( J  e.  (TopOn `  X
)  -> 𝑘Gen  =  ( j  e.  Top  |->  { x  e.  ~P U. j  | 
A. k  e.  ~P  U. j ( ( jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( jt  k ) ) } ) )
3 unieq 3852 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
4 toponuni 16681 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
54eqcomd 2301 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  U. J  =  X )
63, 5sylan9eqr 2350 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  U. j  =  X )
76pweqd 3643 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  ~P U. j  =  ~P X
)
8 oveq1 5881 . . . . . . 7  |-  ( j  =  J  ->  (
jt  k )  =  ( Jt  k ) )
98eleq1d 2362 . . . . . 6  |-  ( j  =  J  ->  (
( jt  k )  e. 
Comp 
<->  ( Jt  k )  e. 
Comp ) )
108eleq2d 2363 . . . . . 6  |-  ( j  =  J  ->  (
( x  i^i  k
)  e.  ( jt  k )  <->  ( x  i^i  k )  e.  ( Jt  k ) ) )
119, 10imbi12d 311 . . . . 5  |-  ( j  =  J  ->  (
( ( jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( jt  k ) )  <-> 
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) ) )
1211adantl 452 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  (
( ( jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( jt  k ) )  <-> 
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) ) )
137, 12raleqbidv 2761 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  ( A. k  e.  ~P  U. j ( ( jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( jt  k ) )  <->  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) ) )
147, 13rabeqbidv 2796 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  { x  e.  ~P U. j  | 
A. k  e.  ~P  U. j ( ( jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( jt  k ) ) }  =  {
x  e.  ~P X  |  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) } )
15 topontop 16680 . 2  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
16 toponmax 16682 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
17 pwexg 4210 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
18 rabexg 4180 . . 3  |-  ( ~P X  e.  _V  ->  { x  e.  ~P X  |  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) }  e.  _V )
1916, 17, 183syl 18 . 2  |-  ( J  e.  (TopOn `  X
)  ->  { x  e.  ~P X  |  A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) }  e.  _V )
202, 14, 15, 19fvmptd 5622 1  |-  ( J  e.  (TopOn `  X
)  ->  (𝑘Gen `  J
)  =  { x  e.  ~P X  |  A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    i^i cin 3164   ~Pcpw 3638   U.cuni 3843    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647  TopOnctopon 16648   Compccmp 17129  𝑘Genckgen 17244
This theorem is referenced by:  elkgen  17247  kgentopon  17249
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-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-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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-top 16652  df-topon 16655  df-kgen 17245
  Copyright terms: Public domain W3C validator