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Theorem kgenval 17490
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 17489 . . 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 11 . 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 3968 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
4 toponuni 16917 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
54eqcomd 2394 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  U. J  =  X )
63, 5sylan9eqr 2443 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  U. j  =  X )
76pweqd 3749 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  ~P U. j  =  ~P X
)
8 oveq1 6029 . . . . . . 7  |-  ( j  =  J  ->  (
jt  k )  =  ( Jt  k ) )
98eleq1d 2455 . . . . . 6  |-  ( j  =  J  ->  (
( jt  k )  e. 
Comp 
<->  ( Jt  k )  e. 
Comp ) )
108eleq2d 2456 . . . . . 6  |-  ( j  =  J  ->  (
( x  i^i  k
)  e.  ( jt  k )  <->  ( x  i^i  k )  e.  ( Jt  k ) ) )
119, 10imbi12d 312 . . . . 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 453 . . . 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 2861 . . 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 2896 . 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 16916 . 2  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
16 toponmax 16918 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
17 pwexg 4326 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
18 rabexg 4296 . . 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 19 . 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 5751 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655   _Vcvv 2901    i^i cin 3264   ~Pcpw 3744   U.cuni 3959    e. cmpt 4209   ` cfv 5396  (class class class)co 6022   ↾t crest 13577   Topctop 16883  TopOnctopon 16884   Compccmp 17373  𝑘Genckgen 17488
This theorem is referenced by:  elkgen  17491  kgentopon  17493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-top 16888  df-topon 16891  df-kgen 17489
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