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Theorem kgenval 17559
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 17558 . . 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 4016 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
4 toponuni 16984 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
54eqcomd 2440 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  U. J  =  X )
63, 5sylan9eqr 2489 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  U. j  =  X )
76pweqd 3796 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  ~P U. j  =  ~P X
)
8 oveq1 6080 . . . . . . 7  |-  ( j  =  J  ->  (
jt  k )  =  ( Jt  k ) )
98eleq1d 2501 . . . . . 6  |-  ( j  =  J  ->  (
( jt  k )  e. 
Comp 
<->  ( Jt  k )  e. 
Comp ) )
108eleq2d 2502 . . . . . 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 2908 . . 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 2943 . 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 16983 . 2  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
16 toponmax 16985 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
17 pwexg 4375 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
18 rabexg 4345 . . 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 5802 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 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948    i^i cin 3311   ~Pcpw 3791   U.cuni 4007    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   ↾t crest 13640   Topctop 16950  TopOnctopon 16951   Compccmp 17441  𝑘Genckgen 17557
This theorem is referenced by:  elkgen  17560  kgentopon  17562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-top 16955  df-topon 16958  df-kgen 17558
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