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Theorem iscusgra 23315
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 4292 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v USGrph  e  <->  V USGrph  E ) )
2 simpl 457 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
3 difeq1 3462 . . . . . 6  |-  ( v  =  V  ->  (
v  \  { k } )  =  ( V  \  { k } ) )
43adantr 465 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v  \  {
k } )  =  ( V  \  {
k } ) )
5 rneq 5060 . . . . . . 7  |-  ( e  =  E  ->  ran  e  =  ran  E )
65adantl 466 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ran  e  =  ran  E )
76eleq2d 2505 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( { n ,  k }  e.  ran  e 
<->  { n ,  k }  e.  ran  E
) )
84, 7raleqbidv 2926 . . . 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 2926 . . 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 23284 . 2  |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v 
A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e ) }
1210, 11brabga 4598 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 1369    e. wcel 1756   A.wral 2710    \ cdif 3320   {csn 3872   {cpr 3874   class class class wbr 4287   ran crn 4836   USGrph cusg 23215   ComplUSGrph ccusgra 23281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-cnv 4843  df-dm 4845  df-rn 4846  df-cusgra 23284
This theorem is referenced by:  iscusgra0  23316  cusgra0v  23319  cusgra1v  23320  cusgra2v  23321  nbcusgra  23322  cusgra3v  23323  cusgraexi  23327  cusgrares  23331  cusgrauvtxb  23355  cusconngra  23513
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