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Theorem catded 25764
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 25761 . . . . 5  |-  Rel  Cat OLD
2 reldcat 25762 . . . . 5  |-  Rel  dom  Cat
OLD
3 relrcat 25763 . . . . 5  |-  Rel  ran  Cat
OLD
41, 2, 33pm3.2i 1130 . . . 4  |-  ( Rel 
Cat OLD  /\  Rel  dom  Cat
OLD  /\  Rel  ran  Cat OLD  )
5 11st22nd 25045 . . . 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 2283 . . . . . 6  |-  ( dom_ `  T )  =  (
dom_ `  T )
87domval 25723 . . . . 5  |-  ( dom_ `  T )  =  ( 1st `  ( 1st `  T ) )
9 eqid 2283 . . . . . 6  |-  ( cod_ `  T )  =  (
cod_ `  T )
109codval 25724 . . . . 5  |-  ( cod_ `  T )  =  ( 2nd `  ( 1st `  T ) )
118, 10opeq12i 3801 . . . 4  |-  <. ( dom_ `  T ) ,  ( cod_ `  T
) >.  =  <. ( 1st `  ( 1st `  T
) ) ,  ( 2nd `  ( 1st `  T ) ) >.
12 eqid 2283 . . . . . 6  |-  ( id_ `  T )  =  ( id_ `  T )
1312idval 25725 . . . . 5  |-  ( id_ `  T )  =  ( 1st `  ( 2nd `  T ) )
14 eqid 2283 . . . . . 6  |-  ( o_
`  T )  =  ( o_ `  T
)
1514cmpval 25726 . . . . 5  |-  ( o_
`  T )  =  ( 2nd `  ( 2nd `  T ) )
1613, 15opeq12i 3801 . . . 4  |-  <. ( id_ `  T ) ,  ( o_ `  T
) >.  =  <. ( 1st `  ( 2nd `  T
) ) ,  ( 2nd `  ( 2nd `  T ) ) >.
1711, 16opeq12i 3801 . . 3  |-  <. <. ( dom_ `  T ) ,  ( cod_ `  T
) >. ,  <. ( id_ `  T ) ,  ( o_ `  T
) >. >.  =  <. <. ( 1st `  ( 1st `  T
) ) ,  ( 2nd `  ( 1st `  T ) ) >. ,  <. ( 1st `  ( 2nd `  T ) ) ,  ( 2nd `  ( 2nd `  T ) )
>. >.
186, 17syl6eqr 2333 . 2  |-  ( T  e.  Cat OLD  ->  T  =  <. <. ( dom_ `  T
) ,  ( cod_ `  T ) >. ,  <. ( id_ `  T ) ,  ( o_ `  T ) >. >. )
19 eqid 2283 . . . . 5  |-  dom  ( dom_ `  T )  =  dom  ( dom_ `  T
)
20 eqid 2283 . . . . 5  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
217, 9, 12, 14, 19, 20cati 25755 . . . 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 2357 1  |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   dom cdm 4689   ran crn 4690   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715   Dedcded 25734    Cat
OLD ccatOLD 25752
This theorem is referenced by:  domc  25765  codc  25766  idc  25767  cmppfc  25768  idosc  25769  cmppfcd  25770  domcmpc  25771  codcmpc  25772  dualcat2  25784  mrdmcd  25794  eqidob  25795  homib  25796  homgrf  25802  idsubidsup  25857
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-eu 2147  df-mo 2148  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-sbc 2992  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-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-catOLD 25753
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