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

Theorem elkgen 17231
Description: Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
elkgen  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  (𝑘Gen `  J )  <->  ( A  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( A  i^i  k
)  e.  ( Jt  k ) ) ) ) )
Distinct variable groups:    A, k    k, J    k, X

Proof of Theorem elkgen
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 kgenval 17230 . . 3  |-  ( 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 ) ) } )
21eleq2d 2350 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  (𝑘Gen `  J )  <->  A  e.  { x  e.  ~P X  |  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) } ) )
3 ineq1 3363 . . . . . . 7  |-  ( x  =  A  ->  (
x  i^i  k )  =  ( A  i^i  k ) )
43eleq1d 2349 . . . . . 6  |-  ( x  =  A  ->  (
( x  i^i  k
)  e.  ( Jt  k )  <->  ( A  i^i  k )  e.  ( Jt  k ) ) )
54imbi2d 307 . . . . 5  |-  ( x  =  A  ->  (
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) )  <-> 
( ( Jt  k )  e.  Comp  ->  ( A  i^i  k )  e.  ( Jt  k ) ) ) )
65ralbidv 2563 . . . 4  |-  ( x  =  A  ->  ( A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) )  <->  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( A  i^i  k )  e.  ( Jt  k ) ) ) )
76elrab 2923 . . 3  |-  ( A  e.  { x  e. 
~P X  |  A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) }  <->  ( A  e. 
~P X  /\  A. k  e.  ~P  X
( ( Jt  k )  e.  Comp  ->  ( A  i^i  k )  e.  ( Jt  k ) ) ) )
8 toponmax 16666 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
9 elpw2g 4174 . . . . 5  |-  ( X  e.  J  ->  ( A  e.  ~P X  <->  A 
C_  X ) )
108, 9syl 15 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  ~P X  <->  A  C_  X
) )
1110anbi1d 685 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( ( A  e.  ~P X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( A  i^i  k )  e.  ( Jt  k ) ) )  <->  ( A  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( A  i^i  k
)  e.  ( Jt  k ) ) ) ) )
127, 11syl5bb 248 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  { x  e.  ~P X  |  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( x  i^i  k
)  e.  ( Jt  k ) ) }  <->  ( A  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( A  i^i  k
)  e.  ( Jt  k ) ) ) ) )
132, 12bitrd 244 1  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  (𝑘Gen `  J )  <->  ( A  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  -> 
( A  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    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   ` cfv 5255  (class class class)co 5858   ↾t crest 13325  TopOnctopon 16632   Compccmp 17113  𝑘Genckgen 17228
This theorem is referenced by:  kgeni  17232  kgentopon  17233  kgenss  17238  kgenidm  17242  iskgen3  17244  kgen2ss  17250  kgencn  17251  kgencn3  17253  txkgen  17346
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
  Copyright terms: Public domain W3C validator