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Theorem strded 25739
Description: Structure of a deductive system. (Contributed by FL, 28-Oct-2007.)
Assertion
Ref Expression
strded  |-  Ded  C_  (
( _V  X.  _V )  X.  ( _V  X.  _V ) )

Proof of Theorem strded
Dummy variables  t 
f  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ded 25735 . 2  |-  Ded  =  { x  |  E. f E. t E. w E. v ( x  = 
<. <. f ,  t
>. ,  <. w ,  v >. >.  /\  ( ( <. <. f ,  t
>. ,  <. w ,  v >. >.  e.  Alg  /\  A. u  e.  dom  w
( ( f `  ( w `  u
) )  =  u  /\  ( t `  ( w `  u
) )  =  u )  /\  A. y  e.  dom  f A. z  e.  dom  f ( <.
z ,  y >.  e.  dom  v  <->  ( f `  z )  =  ( t `  y ) ) )  /\  ( A. y  e.  dom  f A. z  e.  dom  f ( ( f `
 z )  =  ( t `  y
)  ->  ( f `  ( z v y ) )  =  ( f `  y ) )  /\  A. y  e.  dom  f A. z  e.  dom  f ( ( f `  z )  =  ( t `  y )  ->  (
t `  ( z
v y ) )  =  ( t `  z ) ) ) ) ) }
2 stcat 25044 . 2  |-  { x  |  E. f E. t E. w E. v ( x  =  <. <. f ,  t >. ,  <. w ,  v >. >.  /\  (
( <. <. f ,  t
>. ,  <. w ,  v >. >.  e.  Alg  /\  A. u  e.  dom  w
( ( f `  ( w `  u
) )  =  u  /\  ( t `  ( w `  u
) )  =  u )  /\  A. y  e.  dom  f A. z  e.  dom  f ( <.
z ,  y >.  e.  dom  v  <->  ( f `  z )  =  ( t `  y ) ) )  /\  ( A. y  e.  dom  f A. z  e.  dom  f ( ( f `
 z )  =  ( t `  y
)  ->  ( f `  ( z v y ) )  =  ( f `  y ) )  /\  A. y  e.  dom  f A. z  e.  dom  f ( ( f `  z )  =  ( t `  y )  ->  (
t `  ( z
v y ) )  =  ( t `  z ) ) ) ) ) }  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
31, 2eqsstri 3208 1  |-  Ded  C_  (
( _V  X.  _V )  X.  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   _Vcvv 2788    C_ wss 3152   <.cop 3643    X. cxp 4687   dom cdm 4689   ` cfv 5255  (class class class)co 5858    Alg calg 25711   Dedcded 25734
This theorem is referenced by:  relded  25740  reldded  25741  relrded  25742
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-ded 25735
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