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

Theorem kgenval 19108
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 19107 . . 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 4099 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
4 toponuni 18532 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
54eqcomd 2448 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  U. J  =  X )
63, 5sylan9eqr 2497 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  U. j  =  X )
76pweqd 3865 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  j  =  J )  ->  ~P U. j  =  ~P X
)
8 oveq1 6098 . . . . . . 7  |-  ( j  =  J  ->  (
jt  k )  =  ( Jt  k ) )
98eleq1d 2509 . . . . . 6  |-  ( j  =  J  ->  (
( jt  k )  e. 
Comp 
<->  ( Jt  k )  e. 
Comp ) )
108eleq2d 2510 . . . . . 6  |-  ( j  =  J  ->  (
( x  i^i  k
)  e.  ( jt  k )  <->  ( x  i^i  k )  e.  ( Jt  k ) ) )
119, 10imbi12d 320 . . . . 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 466 . . . 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 2931 . . 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 2967 . 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 18531 . 2  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
16 toponmax 18533 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
17 pwexg 4476 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
18 rabexg 4442 . . 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 20 . 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 5779 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719   _Vcvv 2972    i^i cin 3327   ~Pcpw 3860   U.cuni 4091    e. cmpt 4350   ` cfv 5418  (class class class)co 6091   ↾t crest 14359   Topctop 18498  TopOnctopon 18499   Compccmp 18989  𝑘Genckgen 19106
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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-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-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-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-top 18503  df-topon 18506  df-kgen 19107
This theorem is referenced by:  elkgen  19109  kgentopon  19111
  Copyright terms: Public domain W3C validator