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Theorem iscusgra 24129
Description: The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
iscusgra  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ComplUSGrph  E  <->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  {
k } ) { n ,  k }  e.  ran  E ) ) )
Distinct variable groups:    k, V, n    k, E, n
Allowed substitution hints:    X( k, n)    Y( k, n)

Proof of Theorem iscusgra
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 4452 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v USGrph  e  <->  V USGrph  E ) )
2 simpl 457 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
3 difeq1 3615 . . . . . 6  |-  ( v  =  V  ->  (
v  \  { k } )  =  ( V  \  { k } ) )
43adantr 465 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v  \  {
k } )  =  ( V  \  {
k } ) )
5 rneq 5226 . . . . . . 7  |-  ( e  =  E  ->  ran  e  =  ran  E )
65adantl 466 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ran  e  =  ran  E )
76eleq2d 2537 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( { n ,  k }  e.  ran  e 
<->  { n ,  k }  e.  ran  E
) )
84, 7raleqbidv 3072 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( A. n  e.  ( v  \  {
k } ) { n ,  k }  e.  ran  e  <->  A. n  e.  ( V  \  {
k } ) { n ,  k }  e.  ran  E ) )
92, 8raleqbidv 3072 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( A. k  e.  v  A. n  e.  ( v  \  {
k } ) { n ,  k }  e.  ran  e  <->  A. k  e.  V  A. n  e.  ( V  \  {
k } ) { n ,  k }  e.  ran  E ) )
101, 9anbi12d 710 . 2  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( v USGrph  e  /\  A. k  e.  v 
A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e )  <->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  {
k } ) { n ,  k }  e.  ran  E ) ) )
11 df-cusgra 24094 . 2  |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v 
A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e ) }
1210, 11brabga 4761 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ComplUSGrph  E  <->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  {
k } ) { n ,  k }  e.  ran  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    \ cdif 3473   {csn 4027   {cpr 4029   class class class wbr 4447   ran crn 5000   USGrph cusg 24003   ComplUSGrph ccusgra 24091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-cnv 5007  df-dm 5009  df-rn 5010  df-cusgra 24094
This theorem is referenced by:  iscusgra0  24130  cusgra0v  24133  cusgra1v  24134  cusgra2v  24135  nbcusgra  24136  cusgra3v  24137  cusgraexi  24141  cusgrares  24145  cusgrauvtxb  24169  cusconngra  24349
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