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Theorem strcat 25760
Description: Structure of a category. (Contributed by FL, 26-Oct-2007.)
Assertion
Ref Expression
strcat  |-  Cat OLD  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )

Proof of Theorem strcat
Dummy variables  f 
g  t  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-catOLD 25753 . 2  |-  Cat OLD  =  { x  |  E. g E. f E. v E. u ( x  = 
<. <. g ,  f
>. ,  <. v ,  u >. >.  /\  ( ( <. <. g ,  f
>. ,  <. v ,  u >. >.  e.  Ded  /\  A. y  e.  dom  g A. z  e.  dom  g A. w  e.  dom  g ( ( ( g `  w )  =  ( f `  z )  /\  (
g `  z )  =  ( f `  y ) )  -> 
( w u ( z u y ) )  =  ( ( w u z ) u y ) ) )  /\  ( A. t  e.  dom  v A. y  e.  dom  g ( ( f `  y
)  =  t  -> 
( ( v `  t ) u y )  =  y )  /\  A. t  e. 
dom  v A. y  e.  dom  g ( ( g `  y )  =  t  ->  (
y u ( v `
 t ) )  =  y ) ) ) ) }
2 stcat 25044 . 2  |-  { x  |  E. g E. f E. v E. u ( x  =  <. <. g ,  f >. ,  <. v ,  u >. >.  /\  (
( <. <. g ,  f
>. ,  <. v ,  u >. >.  e.  Ded  /\  A. y  e.  dom  g A. z  e.  dom  g A. w  e.  dom  g ( ( ( g `  w )  =  ( f `  z )  /\  (
g `  z )  =  ( f `  y ) )  -> 
( w u ( z u y ) )  =  ( ( w u z ) u y ) ) )  /\  ( A. t  e.  dom  v A. y  e.  dom  g ( ( f `  y
)  =  t  -> 
( ( v `  t ) u y )  =  y )  /\  A. t  e. 
dom  v A. y  e.  dom  g ( ( g `  y )  =  t  ->  (
y u ( v `
 t ) )  =  y ) ) ) ) }  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
31, 2eqsstri 3208 1  |-  Cat OLD  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   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   Dedcded 25734    Cat
OLD ccatOLD 25752
This theorem is referenced by:  relcat  25761  reldcat  25762  relrcat  25763  issubcata  25846
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-catOLD 25753
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