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Theorem frgra0v 25713
Description: Any graph with no vertex is a friendship graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
frgra0v  |-  ( (/) FriendGrph  E  <->  E  =  (/) )

Proof of Theorem frgra0v
Dummy variables  k 
l  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgra 25712 . . 3  |-  ( (/) FriendGrph  E  -> 
(/) USGrph  E )
2 usgra0v 25090 . . 3  |-  ( (/) USGrph  E  <-> 
E  =  (/) )
31, 2sylib 200 . 2  |-  ( (/) FriendGrph  E  ->  E  =  (/) )
42biimpri 210 . . 3  |-  ( E  =  (/)  ->  (/) USGrph  E )
5 ral0 3903 . . . 4  |-  A. k  e.  (/)  A. l  e.  ( (/)  \  { k } ) E! x  e.  (/)  { { x ,  k } ,  { x ,  l } }  C_  ran  E
65a1i 11 . . 3  |-  ( E  =  (/)  ->  A. k  e.  (/)  A. l  e.  ( (/)  \  { k } ) E! x  e.  (/)  { { x ,  k } ,  { x ,  l } }  C_  ran  E )
7 0ex 4554 . . . 4  |-  (/)  e.  _V
8 eleq1 2495 . . . . 5  |-  ( E  =  (/)  ->  ( E  e.  _V  <->  (/)  e.  _V ) )
97, 8mpbiri 237 . . . 4  |-  ( E  =  (/)  ->  E  e. 
_V )
10 isfrgra 25710 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) FriendGrph  E  <-> 
( (/) USGrph  E  /\  A. k  e.  (/)  A. l  e.  ( (/)  \  { k } ) E! x  e.  (/)  { { x ,  k } ,  { x ,  l } }  C_  ran  E ) ) )
117, 9, 10sylancr 668 . . 3  |-  ( E  =  (/)  ->  ( (/) FriendGrph  E  <->  (
(/) USGrph  E  /\  A. k  e.  (/)  A. l  e.  ( (/)  \  { k } ) E! x  e.  (/)  { { x ,  k } ,  { x ,  l } }  C_  ran  E ) ) )
124, 6, 11mpbir2and 931 . 2  |-  ( E  =  (/)  ->  (/) FriendGrph  E )
133, 12impbii 191 1  |-  ( (/) FriendGrph  E  <->  E  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   A.wral 2776   E!wreu 2778   _Vcvv 3082    \ cdif 3434    C_ wss 3437   (/)c0 3762   {csn 3997   {cpr 3999   class class class wbr 4421   ran crn 4852   USGrph cusg 25049   FriendGrph cfrgra 25708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-usgra 25052  df-frgra 25709
This theorem is referenced by:  frgra0  25714
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