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Theorem 3v3e3cycl 25082
Description: If and only if there is a 3-cycle in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
Assertion
Ref Expression
3v3e3cycl  |-  ( V USGrph  E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3 )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) )
Distinct variable groups:    E, a,
b, c, f, p    V, a, b, c, f, p

Proof of Theorem 3v3e3cycl
StepHypRef Expression
1 usgrafun 24766 . . 3  |-  ( V USGrph  E  ->  Fun  E )
2 19.41vv 1796 . . . . 5  |-  ( E. f E. p ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  <->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E
) )
3 simpr 459 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  Fun  E )
4 simpll 752 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  f
( V Cycles  E )
p )
5 simplr 754 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  ( # `
 f )  =  3 )
6 3v3e3cycl1 25061 . . . . . . 7  |-  ( ( Fun  E  /\  f
( V Cycles  E )
p  /\  ( # `  f
)  =  3 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) )
73, 4, 5, 6syl3anc 1230 . . . . . 6  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
87exlimivv 1744 . . . . 5  |-  ( E. f E. p ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
92, 8sylbir 213 . . . 4  |-  ( ( E. f E. p
( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
109expcom 433 . . 3  |-  ( Fun 
E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) ) )
111, 10syl 17 . 2  |-  ( V USGrph  E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) )
12 3v3e3cycl2 25081 . 2  |-  ( V USGrph  E  ->  ( E. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E )  ->  E. f E. p
( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 ) ) )
1311, 12impbid 190 1  |-  ( V USGrph  E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3 )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842   E.wrex 2755   {cpr 3974   class class class wbr 4395   ran crn 4824   Fun wfun 5563   ` cfv 5569  (class class class)co 6278   3c3 10627   #chash 12452   USGrph cusg 24747   Cycles ccycl 24924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-fzo 11855  df-hash 12453  df-word 12591  df-usgra 24750  df-wlk 24925  df-trail 24926  df-pth 24927  df-cycl 24930
This theorem is referenced by: (None)
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