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Theorem ismgra 25813
Description: The predicate "is a directed multi graph". (Contributed by FL, 10-Jan-2008.)
Assertion
Ref Expression
ismgra  |-  ( ( D  e.  A  /\  C  e.  B  /\  U  e.  F )  ->  ( <. <. D ,  C >. ,  U >.  e.  Dgra  <->  ( D : dom  D --> U  /\  C : dom  D --> U ) ) )

Proof of Theorem ismgra
Dummy variables  c 
d  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mgra 25812 . . 3  |-  Dgra  =  { <. <. d ,  c
>. ,  u >.  |  ( d : dom  d
--> u  /\  c : dom  d --> u ) }
21eleq2i 2360 . 2  |-  ( <. <. D ,  C >. ,  U >.  e.  Dgra  <->  <. <. D ,  C >. ,  U >.  e.  { <. <.
d ,  c >. ,  u >.  |  (
d : dom  d --> u  /\  c : dom  d
--> u ) } )
3 feq1 5391 . . . . 5  |-  ( d  =  D  ->  (
d : dom  d --> u 
<->  D : dom  d --> u ) )
4 dmeq 4895 . . . . . 6  |-  ( d  =  D  ->  dom  d  =  dom  D )
54feq2d 5396 . . . . 5  |-  ( d  =  D  ->  ( D : dom  d --> u  <-> 
D : dom  D --> u ) )
63, 5bitrd 244 . . . 4  |-  ( d  =  D  ->  (
d : dom  d --> u 
<->  D : dom  D --> u ) )
74feq2d 5396 . . . 4  |-  ( d  =  D  ->  (
c : dom  d --> u 
<->  c : dom  D --> u ) )
86, 7anbi12d 691 . . 3  |-  ( d  =  D  ->  (
( d : dom  d
--> u  /\  c : dom  d --> u )  <-> 
( D : dom  D --> u  /\  c : dom  D --> u ) ) )
9 feq1 5391 . . . 4  |-  ( c  =  C  ->  (
c : dom  D --> u 
<->  C : dom  D --> u ) )
109anbi2d 684 . . 3  |-  ( c  =  C  ->  (
( D : dom  D --> u  /\  c : dom  D --> u )  <-> 
( D : dom  D --> u  /\  C : dom  D --> u ) ) )
11 feq3 5393 . . . 4  |-  ( u  =  U  ->  ( D : dom  D --> u  <->  D : dom  D --> U ) )
12 feq3 5393 . . . 4  |-  ( u  =  U  ->  ( C : dom  D --> u  <->  C : dom  D --> U ) )
1311, 12anbi12d 691 . . 3  |-  ( u  =  U  ->  (
( D : dom  D --> u  /\  C : dom  D --> u )  <->  ( D : dom  D --> U  /\  C : dom  D --> U ) ) )
148, 10, 13eloprabg 5951 . 2  |-  ( ( D  e.  A  /\  C  e.  B  /\  U  e.  F )  ->  ( <. <. D ,  C >. ,  U >.  e.  { <. <. d ,  c
>. ,  u >.  |  ( d : dom  d
--> u  /\  c : dom  d --> u ) }  <->  ( D : dom  D --> U  /\  C : dom  D --> U ) ) )
152, 14syl5bb 248 1  |-  ( ( D  e.  A  /\  C  e.  B  /\  U  e.  F )  ->  ( <. <. D ,  C >. ,  U >.  e.  Dgra  <->  ( D : dom  D --> U  /\  C : dom  D --> U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656   dom cdm 4705   -->wf 5267   {coprab 5875   Dgracmgra 25811
This theorem is referenced by:  aidm2  25853
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  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-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-oprab 5878  df-mgra 25812
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