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Theorem iscusgra 24873
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 4400 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v USGrph  e  <->  V USGrph  E ) )
2 simpl 455 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
3 difeq1 3554 . . . . . 6  |-  ( v  =  V  ->  (
v  \  { k } )  =  ( V  \  { k } ) )
43adantr 463 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v  \  {
k } )  =  ( V  \  {
k } ) )
5 rneq 5049 . . . . . . 7  |-  ( e  =  E  ->  ran  e  =  ran  E )
65adantl 464 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ran  e  =  ran  E )
76eleq2d 2472 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( { n ,  k }  e.  ran  e 
<->  { n ,  k }  e.  ran  E
) )
84, 7raleqbidv 3018 . . . 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 3018 . . 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 709 . 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 24838 . 2  |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v 
A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e ) }
1210, 11brabga 4704 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 367    = wceq 1405    e. wcel 1842   A.wral 2754    \ cdif 3411   {csn 3972   {cpr 3974   class class class wbr 4395   ran crn 4824   USGrph cusg 24747   ComplUSGrph ccusgra 24835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-cnv 4831  df-dm 4833  df-rn 4834  df-cusgra 24838
This theorem is referenced by:  iscusgra0  24874  cusgra0v  24877  cusgra1v  24878  cusgra2v  24879  nbcusgra  24880  cusgra3v  24881  cusgraexi  24885  cusgrares  24889  cusgrauvtxb  24913  cusconngra  25093
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