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Theorem domc 25868
Description: The 1st "axiom" of a category:  ( dom_ `  T
) is a mapping from the morphisms of  T to the objects of  T. (Contributed by FL, 2-Jan-2008.)
Hypotheses
Ref Expression
cat.1  |-  M  =  dom  D
cat.3  |-  D  =  ( dom_ `  T
)
cat.2  |-  O  =  dom  J
cat.5  |-  J  =  ( id_ `  T
)
Assertion
Ref Expression
domc  |-  ( T  e.  Cat OLD  ->  D : M --> O )

Proof of Theorem domc
StepHypRef Expression
1 catded 25867 . 2  |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
2 dedalg 25846 . 2  |-  ( T  e.  Ded  ->  T  e.  Alg  )
3 cat.1 . . 3  |-  M  =  dom  D
4 cat.3 . . 3  |-  D  =  ( dom_ `  T
)
5 cat.2 . . 3  |-  O  =  dom  J
6 cat.5 . . 3  |-  J  =  ( id_ `  T
)
73, 4, 5, 6doma 25831 . 2  |-  ( T  e.  Alg  ->  D : M --> O )
81, 2, 73syl 18 1  |-  ( T  e.  Cat OLD  ->  D : M --> O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   dom cdm 4705   -->wf 5267   ` cfv 5271    Alg calg 25814   dom_cdom_ 25815   id_cid_ 25817   Dedcded 25837    Cat
OLD ccatOLD 25855
This theorem is referenced by:  dmo  25879  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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