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Theorem catded 25867
Description: A category is a deductive system. (Contributed by FL, 26-Oct-2007.)
Assertion
Ref Expression
catded  |-  ( T  e.  Cat OLD  ->  T  e.  Ded )

Proof of Theorem catded
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcat 25864 . . . . 5  |-  Rel  Cat OLD
2 reldcat 25865 . . . . 5  |-  Rel  dom  Cat
OLD
3 relrcat 25866 . . . . 5  |-  Rel  ran  Cat
OLD
41, 2, 33pm3.2i 1130 . . . 4  |-  ( Rel 
Cat OLD  /\  Rel  dom  Cat
OLD  /\  Rel  ran  Cat OLD  )
5 11st22nd 25148 . . . 4  |-  ( ( ( Rel  Cat OLD  /\ 
Rel  dom  Cat OLD  /\  Rel  ran  Cat OLD  )  /\  T  e.  Cat OLD  )  ->  T  =  <. <. ( 1st `  ( 1st `  T ) ) ,  ( 2nd `  ( 1st `  T ) )
>. ,  <. ( 1st `  ( 2nd `  T
) ) ,  ( 2nd `  ( 2nd `  T ) ) >. >. )
64, 5mpan 651 . . 3  |-  ( T  e.  Cat OLD  ->  T  =  <. <. ( 1st `  ( 1st `  T ) ) ,  ( 2nd `  ( 1st `  T ) )
>. ,  <. ( 1st `  ( 2nd `  T
) ) ,  ( 2nd `  ( 2nd `  T ) ) >. >. )
7 eqid 2296 . . . . . 6  |-  ( dom_ `  T )  =  (
dom_ `  T )
87domval 25826 . . . . 5  |-  ( dom_ `  T )  =  ( 1st `  ( 1st `  T ) )
9 eqid 2296 . . . . . 6  |-  ( cod_ `  T )  =  (
cod_ `  T )
109codval 25827 . . . . 5  |-  ( cod_ `  T )  =  ( 2nd `  ( 1st `  T ) )
118, 10opeq12i 3817 . . . 4  |-  <. ( dom_ `  T ) ,  ( cod_ `  T
) >.  =  <. ( 1st `  ( 1st `  T
) ) ,  ( 2nd `  ( 1st `  T ) ) >.
12 eqid 2296 . . . . . 6  |-  ( id_ `  T )  =  ( id_ `  T )
1312idval 25828 . . . . 5  |-  ( id_ `  T )  =  ( 1st `  ( 2nd `  T ) )
14 eqid 2296 . . . . . 6  |-  ( o_
`  T )  =  ( o_ `  T
)
1514cmpval 25829 . . . . 5  |-  ( o_
`  T )  =  ( 2nd `  ( 2nd `  T ) )
1613, 15opeq12i 3817 . . . 4  |-  <. ( id_ `  T ) ,  ( o_ `  T
) >.  =  <. ( 1st `  ( 2nd `  T
) ) ,  ( 2nd `  ( 2nd `  T ) ) >.
1711, 16opeq12i 3817 . . 3  |-  <. <. ( dom_ `  T ) ,  ( cod_ `  T
) >. ,  <. ( id_ `  T ) ,  ( o_ `  T
) >. >.  =  <. <. ( 1st `  ( 1st `  T
) ) ,  ( 2nd `  ( 1st `  T ) ) >. ,  <. ( 1st `  ( 2nd `  T ) ) ,  ( 2nd `  ( 2nd `  T ) )
>. >.
186, 17syl6eqr 2346 . 2  |-  ( T  e.  Cat OLD  ->  T  =  <. <. ( dom_ `  T
) ,  ( cod_ `  T ) >. ,  <. ( id_ `  T ) ,  ( o_ `  T ) >. >. )
19 eqid 2296 . . . . 5  |-  dom  ( dom_ `  T )  =  dom  ( dom_ `  T
)
20 eqid 2296 . . . . 5  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
217, 9, 12, 14, 19, 20cati 25858 . . . 4  |-  ( T  e.  Cat OLD  ->  ( ( <. <. ( dom_ `  T
) ,  ( cod_ `  T ) >. ,  <. ( id_ `  T ) ,  ( o_ `  T ) >. >.  e.  Ded  /\ 
A. x  e.  dom  ( dom_ `  T ) A. y  e.  dom  ( dom_ `  T ) A. z  e.  dom  ( dom_ `  T )
( ( ( (
dom_ `  T ) `  z )  =  ( ( cod_ `  T
) `  y )  /\  ( ( dom_ `  T
) `  y )  =  ( ( cod_ `  T ) `  x
) )  ->  (
z ( o_ `  T ) ( y ( o_ `  T
) x ) )  =  ( ( z ( o_ `  T
) y ) ( o_ `  T ) x ) ) )  /\  ( A. w  e.  dom  ( id_ `  T
) A. x  e. 
dom  ( dom_ `  T
) ( ( (
cod_ `  T ) `  x )  =  w  ->  ( ( ( id_ `  T ) `
 w ) ( o_ `  T ) x )  =  x )  /\  A. w  e.  dom  ( id_ `  T
) A. x  e. 
dom  ( dom_ `  T
) ( ( (
dom_ `  T ) `  x )  =  w  ->  ( x ( o_ `  T ) ( ( id_ `  T
) `  w )
)  =  x ) ) ) )
2221simpld 445 . . 3  |-  ( T  e.  Cat OLD  ->  (
<. <. ( dom_ `  T
) ,  ( cod_ `  T ) >. ,  <. ( id_ `  T ) ,  ( o_ `  T ) >. >.  e.  Ded  /\ 
A. x  e.  dom  ( dom_ `  T ) A. y  e.  dom  ( dom_ `  T ) A. z  e.  dom  ( dom_ `  T )
( ( ( (
dom_ `  T ) `  z )  =  ( ( cod_ `  T
) `  y )  /\  ( ( dom_ `  T
) `  y )  =  ( ( cod_ `  T ) `  x
) )  ->  (
z ( o_ `  T ) ( y ( o_ `  T
) x ) )  =  ( ( z ( o_ `  T
) y ) ( o_ `  T ) x ) ) ) )
2322simpld 445 . 2  |-  ( T  e.  Cat OLD  ->  <. <. ( dom_ `  T
) ,  ( cod_ `  T ) >. ,  <. ( id_ `  T ) ,  ( o_ `  T ) >. >.  e.  Ded )
2418, 23eqeltrd 2370 1  |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   dom cdm 4705   ran crn 4706   Rel wrel 4710   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   o_co_ 25818   Dedcded 25837    Cat
OLD ccatOLD 25855
This theorem is referenced by:  domc  25868  codc  25869  idc  25870  cmppfc  25871  idosc  25872  cmppfcd  25873  domcmpc  25874  codcmpc  25875  dualcat2  25887  mrdmcd  25897  eqidob  25898  homib  25899  homgrf  25905  idsubidsup  25960
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-eu 2160  df-mo 2161  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-sbc 3005  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-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-catOLD 25856
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