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Theorem iscusgra 21418
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 4177 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v USGrph  e  <->  V USGrph  E ) )
2 simpl 444 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
3 difeq1 3418 . . . . . 6  |-  ( v  =  V  ->  (
v  \  { k } )  =  ( V  \  { k } ) )
43adantr 452 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v  \  {
k } )  =  ( V  \  {
k } ) )
5 rneq 5054 . . . . . . 7  |-  ( e  =  E  ->  ran  e  =  ran  E )
65adantl 453 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ran  e  =  ran  E )
76eleq2d 2471 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( { n ,  k }  e.  ran  e 
<->  { n ,  k }  e.  ran  E
) )
84, 7raleqbidv 2876 . . . 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 2876 . . 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 692 . 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 21387 . 2  |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v 
A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e ) }
1210, 11brabga 4429 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666    \ cdif 3277   {csn 3774   {cpr 3775   class class class wbr 4172   ran crn 4838   USGrph cusg 21318   ComplUSGrph ccusgra 21384
This theorem is referenced by:  iscusgra0  21419  cusgra0v  21422  cusgra1v  21423  cusgra2v  21424  nbcusgra  21425  cusgra3v  21426  cusgraexi  21430  cusgrares  21434  cusgrauvtxb  21458  cusconngra  21616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-cnv 4845  df-dm 4847  df-rn 4848  df-cusgra 21387
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