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Theorem strded 25842
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 25838 . 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 25147 . 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 3221 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   _Vcvv 2801    C_ wss 3165   <.cop 3656    X. cxp 4703   dom cdm 4705   ` cfv 5271  (class class class)co 5874    Alg calg 25814   Dedcded 25837
This theorem is referenced by:  relded  25843  reldded  25844  relrded  25845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-ded 25838
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