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

Theorem elkgen 20203
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 20202 . . 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 2524 . 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 3679 . . . . . . 7  |-  ( x  =  A  ->  (
x  i^i  k )  =  ( A  i^i  k ) )
43eleq1d 2523 . . . . . 6  |-  ( x  =  A  ->  (
( x  i^i  k
)  e.  ( Jt  k )  <->  ( A  i^i  k )  e.  ( Jt  k ) ) )
54imbi2d 314 . . . . 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 2893 . . . 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 3254 . . 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 19596 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
9 elpw2g 4600 . . . . 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 702 . . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   ` cfv 5570  (class class class)co 6270   ↾t crest 14910  TopOnctopon 19562   Compccmp 20053  𝑘Genckgen 20200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-top 19566  df-topon 19569  df-kgen 20201
This theorem is referenced by:  kgeni  20204  kgentopon  20205  kgenss  20210  kgenidm  20214  iskgen3  20216  kgen2ss  20222  kgencn  20223  kgencn3  20225  txkgen  20319
  Copyright terms: Public domain W3C validator