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Theorem 3v3e3cycl 21605
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 21331 . . 3  |-  ( V USGrph  E  ->  Fun  E )
2 19.41vv 1921 . . . . 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 448 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  Fun  E )
4 simpll 731 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  f
( V Cycles  E )
p )
5 simplr 732 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  ( # `
 f )  =  3 )
6 3v3e3cycl1 21584 . . . . . . 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 1184 . . . . . 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 1642 . . . . 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 205 . . . 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 425 . . 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 16 . 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 21604 . 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 184 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   E.wrex 2667   {cpr 3775   class class class wbr 4172   ran crn 4838   Fun wfun 5407   ` cfv 5413  (class class class)co 6040   3c3 10006   #chash 11573   USGrph cusg 21318   Cycles ccycl 21468
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-usgra 21320  df-wlk 21469  df-trail 21470  df-pth 21471  df-cycl 21474
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