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

Theorem elkgen 19236
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 19235 . . 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 2522 . 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 3648 . . . . . . 7  |-  ( x  =  A  ->  (
x  i^i  k )  =  ( A  i^i  k ) )
43eleq1d 2521 . . . . . 6  |-  ( x  =  A  ->  (
( x  i^i  k
)  e.  ( Jt  k )  <->  ( A  i^i  k )  e.  ( Jt  k ) ) )
54imbi2d 316 . . . . 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 2843 . . . 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 3218 . . 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 18660 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
9 elpw2g 4558 . . . . 5  |-  ( X  e.  J  ->  ( A  e.  ~P X  <->  A 
C_  X ) )
108, 9syl 16 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( A  e.  ~P X  <->  A  C_  X
) )
1110anbi1d 704 . . 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 257 . 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 253 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796   {crab 2800    i^i cin 3430    C_ wss 3431   ~Pcpw 3963   ` cfv 5521  (class class class)co 6195   ↾t crest 14473  TopOnctopon 18626   Compccmp 19116  𝑘Genckgen 19233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-ov 6198  df-top 18630  df-topon 18633  df-kgen 19234
This theorem is referenced by:  kgeni  19237  kgentopon  19238  kgenss  19243  kgenidm  19247  iskgen3  19249  kgen2ss  19255  kgencn  19256  kgencn3  19258  txkgen  19352
  Copyright terms: Public domain W3C validator