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Theorem morcat 25883
Description: Two ways to define the set of the morphisms of a category. (Contributed by FL, 19-Sep-2009.)
Assertion
Ref Expression
morcat  |-  ( T  e.  Cat OLD  ->  dom  ( dom_ `  T
)  =  dom  ( cod_ `  T ) )

Proof of Theorem morcat
StepHypRef Expression
1 eqid 2296 . . 3  |-  dom  ( dom_ `  T )  =  dom  ( dom_ `  T
)
2 eqid 2296 . . 3  |-  ( dom_ `  T )  =  (
dom_ `  T )
3 eqid 2296 . . 3  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
4 eqid 2296 . . 3  |-  ( id_ `  T )  =  ( id_ `  T )
5 eqid 2296 . . 3  |-  ( cod_ `  T )  =  (
cod_ `  T )
61, 2, 3, 4, 5codc 25869 . 2  |-  ( T  e.  Cat OLD  ->  (
cod_ `  T ) : dom  ( dom_ `  T
) --> dom  ( id_ `  T ) )
7 fdm 5409 . . 3  |-  ( (
cod_ `  T ) : dom  ( dom_ `  T
) --> dom  ( id_ `  T )  ->  dom  ( cod_ `  T )  =  dom  ( dom_ `  T
) )
87eqcomd 2301 . 2  |-  ( (
cod_ `  T ) : dom  ( dom_ `  T
) --> dom  ( id_ `  T )  ->  dom  ( dom_ `  T )  =  dom  ( cod_ `  T
) )
96, 8syl 15 1  |-  ( T  e.  Cat OLD  ->  dom  ( dom_ `  T
)  =  dom  ( cod_ `  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   dom cdm 4705   -->wf 5267   ` cfv 5271   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817    Cat
OLD ccatOLD 25855
This theorem is referenced by:  idsubfun  25961
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-alg 25819  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-ded 25838  df-catOLD 25856
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