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Theorem iscusgra 23515
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 4404 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v USGrph  e  <->  V USGrph  E ) )
2 simpl 457 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
3 difeq1 3574 . . . . . 6  |-  ( v  =  V  ->  (
v  \  { k } )  =  ( V  \  { k } ) )
43adantr 465 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v  \  {
k } )  =  ( V  \  {
k } ) )
5 rneq 5172 . . . . . . 7  |-  ( e  =  E  ->  ran  e  =  ran  E )
65adantl 466 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ran  e  =  ran  E )
76eleq2d 2524 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( { n ,  k }  e.  ran  e 
<->  { n ,  k }  e.  ran  E
) )
84, 7raleqbidv 3035 . . . 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 3035 . . 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 23484 . 2  |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v 
A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e ) }
1210, 11brabga 4710 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 1370    e. wcel 1758   A.wral 2798    \ cdif 3432   {csn 3984   {cpr 3986   class class class wbr 4399   ran crn 4948   USGrph cusg 23415   ComplUSGrph ccusgra 23481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-cnv 4955  df-dm 4957  df-rn 4958  df-cusgra 23484
This theorem is referenced by:  iscusgra0  23516  cusgra0v  23519  cusgra1v  23520  cusgra2v  23521  nbcusgra  23522  cusgra3v  23523  cusgraexi  23527  cusgrares  23531  cusgrauvtxb  23555  cusconngra  23713
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