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Theorem domcmpc 25771
Description: The 7th "axiom" of a category: when  ( G ( o_ `  T ) F ) is defined its domain is the domain of 
F. (Contributed by FL, 10-Mar-2008.)
Hypotheses
Ref Expression
cmppfcd.1  |-  M  =  dom  D
cmppfcd.2  |-  D  =  ( dom_ `  T
)
cmppfcd.3  |-  C  =  ( cod_ `  T
)
cmppfcd.4  |-  R  =  ( o_ `  T
)
Assertion
Ref Expression
domcmpc  |-  ( ( T  e.  Cat OLD  /\  F  e.  M  /\  G  e.  M )  ->  ( ( D `  G )  =  ( C `  F )  ->  ( D `  ( G R F ) )  =  ( D `
 F ) ) )

Proof of Theorem domcmpc
StepHypRef Expression
1 catded 25764 . 2  |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
2 cmppfcd.1 . . 3  |-  M  =  dom  D
3 cmppfcd.2 . . 3  |-  D  =  ( dom_ `  T
)
4 cmppfcd.3 . . 3  |-  C  =  ( cod_ `  T
)
5 cmppfcd.4 . . 3  |-  R  =  ( o_ `  T
)
62, 3, 4, 5domcmpd 25746 . 2  |-  ( ( T  e.  Ded  /\  F  e.  M  /\  G  e.  M )  ->  ( ( D `  G )  =  ( C `  F )  ->  ( D `  ( G R F ) )  =  ( D `
 F ) ) )
71, 6syl3an1 1215 1  |-  ( ( T  e.  Cat OLD  /\  F  e.  M  /\  G  e.  M )  ->  ( ( D `  G )  =  ( C `  F )  ->  ( D `  ( G R F ) )  =  ( D `
 F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   dom cdm 4689   ` cfv 5255  (class class class)co 5858   dom_cdom_ 25712   cod_ccod_ 25713   o_co_ 25715   Dedcded 25734    Cat
OLD ccatOLD 25752
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-ded 25735  df-catOLD 25753
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