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Theorem cmpmorp 25779
Description: Conditions for a composite to be a morphism. (Contributed by FL, 10-Mar-2008.)
Hypotheses
Ref Expression
cmpmorp.1  |-  M  =  dom  ( dom_ `  T
)
cmpmorp.2  |-  D  =  ( dom_ `  T
)
cmpmorp.3  |-  C  =  ( cod_ `  T
)
cmpmorp.4  |-  R  =  ( o_ `  T
)
Assertion
Ref Expression
cmpmorp  |-  ( ( T  e.  Cat OLD  /\  F  e.  M  /\  G  e.  M )  ->  ( ( D `  G )  =  ( C `  F )  ->  ( G R F )  e.  M
) )

Proof of Theorem cmpmorp
StepHypRef Expression
1 cmpmorp.1 . . . . . . . . 9  |-  M  =  dom  ( dom_ `  T
)
2 eqid 2283 . . . . . . . . 9  |-  ( dom_ `  T )  =  (
dom_ `  T )
3 cmpmorp.4 . . . . . . . . 9  |-  R  =  ( o_ `  T
)
41, 2, 3cmppfc 25768 . . . . . . . 8  |-  ( T  e.  Cat OLD  ->  ( Fun  R  /\  dom  R 
C_  ( M  X.  M )  /\  ran  R 
C_  M ) )
543ad2ant1 976 . . . . . . 7  |-  ( ( T  e.  Cat OLD  /\  F  e.  M  /\  G  e.  M )  ->  ( Fun  R  /\  dom  R  C_  ( M  X.  M )  /\  ran  R 
C_  M ) )
65simp1d 967 . . . . . 6  |-  ( ( T  e.  Cat OLD  /\  F  e.  M  /\  G  e.  M )  ->  Fun  R )
76adantr 451 . . . . 5  |-  ( ( ( T  e.  Cat OLD 
/\  F  e.  M  /\  G  e.  M
)  /\  ( D `  G )  =  ( C `  F ) )  ->  Fun  R )
8 cmpmorp.2 . . . . . . . . . 10  |-  D  =  ( dom_ `  T
)
98eqcomi 2287 . . . . . . . . 9  |-  ( dom_ `  T )  =  D
109dmeqi 4880 . . . . . . . 8  |-  dom  ( dom_ `  T )  =  dom  D
111, 10eqtri 2303 . . . . . . 7  |-  M  =  dom  D
12 cmpmorp.3 . . . . . . 7  |-  C  =  ( cod_ `  T
)
1311, 8, 12, 3cmppfcd 25770 . . . . . 6  |-  ( ( T  e.  Cat OLD  /\  F  e.  M  /\  G  e.  M )  ->  ( <. G ,  F >.  e.  dom  R  <->  ( D `  G )  =  ( C `  F ) ) )
1413biimpar 471 . . . . 5  |-  ( ( ( T  e.  Cat OLD 
/\  F  e.  M  /\  G  e.  M
)  /\  ( D `  G )  =  ( C `  F ) )  ->  <. G ,  F >.  e.  dom  R
)
157, 14jca 518 . . . 4  |-  ( ( ( T  e.  Cat OLD 
/\  F  e.  M  /\  G  e.  M
)  /\  ( D `  G )  =  ( C `  F ) )  ->  ( Fun  R  /\  <. G ,  F >.  e.  dom  R ) )
1615ex 423 . . 3  |-  ( ( T  e.  Cat OLD  /\  F  e.  M  /\  G  e.  M )  ->  ( ( D `  G )  =  ( C `  F )  ->  ( Fun  R  /\  <. G ,  F >.  e.  dom  R ) ) )
17 fnovrn2 25050 . . 3  |-  ( ( Fun  R  /\  <. G ,  F >.  e.  dom  R )  ->  ( G R F )  e.  ran  R )
1816, 17syl6 29 . 2  |-  ( ( T  e.  Cat OLD  /\  F  e.  M  /\  G  e.  M )  ->  ( ( D `  G )  =  ( C `  F )  ->  ( G R F )  e.  ran  R ) )
195simp3d 969 . . 3  |-  ( ( T  e.  Cat OLD  /\  F  e.  M  /\  G  e.  M )  ->  ran  R  C_  M
)
2019sseld 3179 . 2  |-  ( ( T  e.  Cat OLD  /\  F  e.  M  /\  G  e.  M )  ->  ( ( G R F )  e.  ran  R  ->  ( G R F )  e.  M
) )
2118, 20syld 40 1  |-  ( ( T  e.  Cat OLD  /\  F  e.  M  /\  G  e.  M )  ->  ( ( D `  G )  =  ( C `  F )  ->  ( G R F )  e.  M
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   <.cop 3643    X. cxp 4687   dom cdm 4689   ran crn 4690   Fun wfun 5249   ` cfv 5255  (class class class)co 5858   dom_cdom_ 25712   cod_ccod_ 25713   o_co_ 25715    Cat
OLD ccatOLD 25752
This theorem is referenced by:  homgrf  25802  idfisf  25841  idsubfun  25858
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-alg 25716  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-ded 25735  df-catOLD 25753
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