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Theorem kgenval 17230
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 17229 . . 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 3836 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
4 toponuni 16665 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
54eqcomd 2288 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  U. J  =  X )
63, 5sylan9eqr 2337 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  U. j  =  X )
76pweqd 3630 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  ~P U. j  =  ~P X
)
8 oveq1 5865 . . . . . . 7  |-  ( j  =  J  ->  (
jt  k )  =  ( Jt  k ) )
98eleq1d 2349 . . . . . 6  |-  ( j  =  J  ->  (
( jt  k )  e. 
Comp 
<->  ( Jt  k )  e. 
Comp ) )
108eleq2d 2350 . . . . . 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 2748 . . 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 2783 . 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 16664 . 2  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
16 toponmax 16666 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
17 pwexg 4194 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
18 rabexg 4164 . . 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 5606 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 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    i^i cin 3151   ~Pcpw 3625   U.cuni 3827    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631  TopOnctopon 16632   Compccmp 17113  𝑘Genckgen 17228
This theorem is referenced by:  elkgen  17231  kgentopon  17233
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-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-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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-top 16636  df-topon 16639  df-kgen 17229
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