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Theorem doma 25728
Description:  (
dom_ `  T ) is a mapping from the morphisms of  T to the objects of  T. (Contributed by FL, 24-Oct-2007.)
Hypotheses
Ref Expression
alg.1  |-  M  =  dom  D
alg.3  |-  D  =  ( dom_ `  T
)
alg.2  |-  O  =  dom  J
alg.5  |-  J  =  ( id_ `  T
)
Assertion
Ref Expression
doma  |-  ( T  e.  Alg  ->  D : M --> O )

Proof of Theorem doma
StepHypRef Expression
1 alg.3 . . . 4  |-  D  =  ( dom_ `  T
)
2 eqid 2283 . . . 4  |-  ( cod_ `  T )  =  (
cod_ `  T )
3 alg.5 . . . 4  |-  J  =  ( id_ `  T
)
4 eqid 2283 . . . 4  |-  ( o_
`  T )  =  ( o_ `  T
)
5 alg.1 . . . 4  |-  M  =  dom  D
6 alg.2 . . . 4  |-  O  =  dom  J
71, 2, 3, 4, 5, 6algi 25727 . . 3  |-  ( T  e.  Alg  ->  (
( D : M --> O  /\  ( cod_ `  T
) : M --> O  /\  J : O --> M )  /\  ( Fun  (
o_ `  T )  /\  dom  ( o_ `  T )  C_  ( M  X.  M )  /\  ran  ( o_ `  T
)  C_  M )
) )
87simpld 445 . 2  |-  ( T  e.  Alg  ->  ( D : M --> O  /\  ( cod_ `  T ) : M --> O  /\  J : O --> M ) )
98simp1d 967 1  |-  ( T  e.  Alg  ->  D : M --> O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152    X. cxp 4687   dom cdm 4689   ran crn 4690   Fun wfun 5249   -->wf 5251   ` cfv 5255    Alg calg 25711   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715
This theorem is referenced by:  rdmob  25748  aidm2  25750  dmrngcmp  25751  domc  25765  dualalg  25782  mrdmcd  25794
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-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-1st 6122  df-2nd 6123  df-alg 25716  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720
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