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Theorem frgr0v 27125
Description: Any null graph (set with no vertices) is a friendship graph iff its edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
Assertion
Ref Expression
frgr0v |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( G e. FriendGraph <-> (iEdg ` G ) = (/) ) )
Proof of Theorem frgr0v
Dummy variables k l x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2622 . . 3 |- (Vtx ` G ) = (Vtx ` G )
2 eqid 2622 . . 3 |- (Edg ` G ) = (Edg ` G )
31, 2frgrusgrfrcond 27123 . 2 |- ( G e. FriendGraph <-> ( G e. USGraph /\ A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) ) )
4 usgruhgr 26078 . . . . 5 |- ( G e. USGraph -> G e. UHGraph )
54adantr 481 . . . 4 |- ( ( G e. USGraph /\ A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) ) -> G e. UHGraph )
6 uhgr0vb 25967 . . . 4 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( G e. UHGraph <-> (iEdg ` G ) = (/) ) )
75, 6syl5ib 234 . . 3 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( ( G e. USGraph /\ A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) ) -> (iEdg ` G ) = (/) ) )
8 simpll 790 . . . . . 6 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) ) /\ (iEdg ` G ) = (/) ) -> G e. W )
9 simpr 477 . . . . . 6 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) ) /\ (iEdg ` G ) = (/) ) -> (iEdg ` G ) = (/) )
108, 9usgr0e 26128 . . . . 5 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) ) /\ (iEdg ` G ) = (/) ) -> G e. USGraph )
11 ral0 4076 . . . . . . 7 |- A. k e. (/) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G )
12 raleq 3138 . . . . . . . 8 |- ( (Vtx ` G ) = (/) -> ( A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) <-> A. k e. (/) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) ) )
1312adantl 482 . . . . . . 7 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) <-> A. k e. (/) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) ) )
1411, 13mpbiri 248 . . . . . 6 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) )
1514adantr 481 . . . . 5 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) ) /\ (iEdg ` G ) = (/) ) -> A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) )
1610, 15jca 554 . . . 4 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) ) /\ (iEdg ` G ) = (/) ) -> ( G e. USGraph /\ A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) ) )
1716ex 450 . . 3 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( (iEdg ` G ) = (/) -> ( G e. USGraph /\ A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) ) ) )
187, 17impbid 202 . 2 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( ( G e. USGraph /\ A. k e. (Vtx ` G ) A. l e. ( (Vtx ` G ) \ { k } ) E! x e. (Vtx ` G ) { { x , k } , { x , l } } C_ (Edg ` G ) ) <-> (iEdg ` G ) = (/) ) )
193, 18syl5bb 272 1 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( G e. FriendGraph <-> (iEdg ` G ) = (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   USGraph cusgr 26044   FriendGraph cfrgr 27120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-2 11079  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-usgr 26046  df-frgr 27121
This theorem is referenced by:  frgr0vb  27126
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