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Theorem aidm2 25750
Description: The underlying directed multi graph of a deductive system. (Contributed by FL, 20-Sep-2009.)
Hypotheses
Ref Expression
aidm2.1  |-  D  =  ( dom_ `  T
)
aidm2.2  |-  C  =  ( cod_ `  T
)
Assertion
Ref Expression
aidm2  |-  ( T  e.  Ded  ->  <. <. D ,  C >. ,  ran  D >.  e.  Dgra )

Proof of Theorem aidm2
StepHypRef Expression
1 dedalg 25743 . . . 4  |-  ( T  e.  Ded  ->  T  e.  Alg  )
2 eqid 2283 . . . . 5  |-  dom  D  =  dom  D
3 aidm2.1 . . . . 5  |-  D  =  ( dom_ `  T
)
4 eqid 2283 . . . . 5  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
5 eqid 2283 . . . . 5  |-  ( id_ `  T )  =  ( id_ `  T )
62, 3, 4, 5doma 25728 . . . 4  |-  ( T  e.  Alg  ->  D : dom  D --> dom  ( id_ `  T ) )
71, 6syl 15 . . 3  |-  ( T  e.  Ded  ->  D : dom  D --> dom  ( id_ `  T ) )
84, 3rdmob 25748 . . . 4  |-  ( T  e.  Ded  ->  ran  D  =  dom  ( id_ `  T ) )
9 feq3 5377 . . . 4  |-  ( ran 
D  =  dom  ( id_ `  T )  -> 
( D : dom  D --> ran  D  <->  D : dom  D --> dom  ( id_ `  T ) ) )
108, 9syl 15 . . 3  |-  ( T  e.  Ded  ->  ( D : dom  D --> ran  D  <->  D : dom  D --> dom  ( id_ `  T ) ) )
117, 10mpbird 223 . 2  |-  ( T  e.  Ded  ->  D : dom  D --> ran  D
)
123dmeqi 4880 . . . . 5  |-  dom  D  =  dom  ( dom_ `  T
)
13 eqid 2283 . . . . 5  |-  ( dom_ `  T )  =  (
dom_ `  T )
14 aidm2.2 . . . . 5  |-  C  =  ( cod_ `  T
)
1512, 13, 4, 5, 14coda 25729 . . . 4  |-  ( T  e.  Alg  ->  C : dom  D --> dom  ( id_ `  T ) )
161, 15syl 15 . . 3  |-  ( T  e.  Ded  ->  C : dom  D --> dom  ( id_ `  T ) )
17 feq3 5377 . . . 4  |-  ( ran 
D  =  dom  ( id_ `  T )  -> 
( C : dom  D --> ran  D  <->  C : dom  D --> dom  ( id_ `  T ) ) )
188, 17syl 15 . . 3  |-  ( T  e.  Ded  ->  ( C : dom  D --> ran  D  <->  C : dom  D --> dom  ( id_ `  T ) ) )
1916, 18mpbird 223 . 2  |-  ( T  e.  Ded  ->  C : dom  D --> ran  D
)
203a1i 10 . . . 4  |-  ( T  e.  Ded  ->  D  =  ( dom_ `  T
) )
21 fvex 5539 . . . 4  |-  ( dom_ `  T )  e.  _V
2220, 21syl6eqel 2371 . . 3  |-  ( T  e.  Ded  ->  D  e.  _V )
2314a1i 10 . . . 4  |-  ( T  e.  Ded  ->  C  =  ( cod_ `  T
) )
24 fvex 5539 . . . 4  |-  ( cod_ `  T )  e.  _V
2523, 24syl6eqel 2371 . . 3  |-  ( T  e.  Ded  ->  C  e.  _V )
2620rneqd 4906 . . . 4  |-  ( T  e.  Ded  ->  ran  D  =  ran  ( dom_ `  T ) )
27 rnexg 4940 . . . . 5  |-  ( (
dom_ `  T )  e.  _V  ->  ran  ( dom_ `  T )  e.  _V )
2821, 27mp1i 11 . . . 4  |-  ( T  e.  Ded  ->  ran  ( dom_ `  T )  e.  _V )
2926, 28eqeltrd 2357 . . 3  |-  ( T  e.  Ded  ->  ran  D  e.  _V )
30 ismgra 25710 . . 3  |-  ( ( D  e.  _V  /\  C  e.  _V  /\  ran  D  e.  _V )  -> 
( <. <. D ,  C >. ,  ran  D >.  e. 
Dgra 
<->  ( D : dom  D --> ran  D  /\  C : dom  D --> ran  D
) ) )
3122, 25, 29, 30syl3anc 1182 . 2  |-  ( T  e.  Ded  ->  ( <. <. D ,  C >. ,  ran  D >.  e. 
Dgra 
<->  ( D : dom  D --> ran  D  /\  C : dom  D --> ran  D
) ) )
3211, 19, 31mpbir2and 888 1  |-  ( T  e.  Ded  ->  <. <. D ,  C >. ,  ran  D >.  e.  Dgra )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   dom cdm 4689   ran crn 4690   -->wf 5251   ` cfv 5255   Dgracmgra 25708    Alg calg 25711   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   Dedcded 25734
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-oprab 5862  df-1st 6122  df-2nd 6123  df-mgra 25709  df-alg 25716  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-ded 25735
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