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Theorem 3cyclfrgra 27770
Description: Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
Assertion
Ref Expression
3cyclfrgra  |-  ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  ->  A. v  e.  V  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3  /\  ( p `  0
)  =  v ) )
Distinct variable groups:    v, E, f, p    v, V, f, p

Proof of Theorem 3cyclfrgra
Dummy variables  b 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3cyclfrgrarn 27768 . 2  |-  ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  ->  A. v  e.  V  E. b  e.  V  E. c  e.  V  ( {
v ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  v }  e.  ran  E ) )
2 frisusgra 27747 . . . . . . 7  |-  ( V FriendGrph  E  ->  V USGrph  E )
32ad4antr 713 . . . . . 6  |-  ( ( ( ( ( V FriendGrph  E  /\  1  <  ( # `
 V ) )  /\  v  e.  V
)  /\  ( b  e.  V  /\  c  e.  V ) )  /\  ( { v ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  v }  e.  ran  E
) )  ->  V USGrph  E )
4 simpr 448 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  /\  v  e.  V )  ->  v  e.  V )
54adantr 452 . . . . . . . 8  |-  ( ( ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  /\  v  e.  V )  /\  (
b  e.  V  /\  c  e.  V )
)  ->  v  e.  V )
6 simprl 733 . . . . . . . 8  |-  ( ( ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  /\  v  e.  V )  /\  (
b  e.  V  /\  c  e.  V )
)  ->  b  e.  V )
7 simprr 734 . . . . . . . 8  |-  ( ( ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  /\  v  e.  V )  /\  (
b  e.  V  /\  c  e.  V )
)  ->  c  e.  V )
85, 6, 73jca 1134 . . . . . . 7  |-  ( ( ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  /\  v  e.  V )  /\  (
b  e.  V  /\  c  e.  V )
)  ->  ( v  e.  V  /\  b  e.  V  /\  c  e.  V ) )
98adantr 452 . . . . . 6  |-  ( ( ( ( ( V FriendGrph  E  /\  1  <  ( # `
 V ) )  /\  v  e.  V
)  /\  ( b  e.  V  /\  c  e.  V ) )  /\  ( { v ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  v }  e.  ran  E
) )  ->  (
v  e.  V  /\  b  e.  V  /\  c  e.  V )
)
10 simpr 448 . . . . . 6  |-  ( ( ( ( ( V FriendGrph  E  /\  1  <  ( # `
 V ) )  /\  v  e.  V
)  /\  ( b  e.  V  /\  c  e.  V ) )  /\  ( { v ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  v }  e.  ran  E
) )  ->  ( { v ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  v }  e.  ran  E
) )
11 constr3cyclpe 21500 . . . . . 6  |-  ( ( V USGrph  E  /\  (
v  e.  V  /\  b  e.  V  /\  c  e.  V )  /\  ( { v ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  v }  e.  ran  E ) )  ->  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3  /\  ( p `  0
)  =  v ) )
123, 9, 10, 11syl3anc 1184 . . . . 5  |-  ( ( ( ( ( V FriendGrph  E  /\  1  <  ( # `
 V ) )  /\  v  e.  V
)  /\  ( b  e.  V  /\  c  e.  V ) )  /\  ( { v ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  v }  e.  ran  E
) )  ->  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3  /\  ( p `  0
)  =  v ) )
1312ex 424 . . . 4  |-  ( ( ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  /\  v  e.  V )  /\  (
b  e.  V  /\  c  e.  V )
)  ->  ( ( { v ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  v }  e.  ran  E
)  ->  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3  /\  ( p `  0
)  =  v ) ) )
1413rexlimdvva 2782 . . 3  |-  ( ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  /\  v  e.  V )  ->  ( E. b  e.  V  E. c  e.  V  ( { v ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  v }  e.  ran  E
)  ->  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3  /\  ( p `  0
)  =  v ) ) )
1514ralimdva 2729 . 2  |-  ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  ->  ( A. v  e.  V  E. b  e.  V  E. c  e.  V  ( { v ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  v }  e.  ran  E
)  ->  A. v  e.  V  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3  /\  ( p `  0
)  =  v ) ) )
161, 15mpd 15 1  |-  ( ( V FriendGrph  E  /\  1  <  ( # `  V
) )  ->  A. v  e.  V  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3  /\  ( p `  0
)  =  v ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652   {cpr 3760   class class class wbr 4155   ran crn 4821   ` cfv 5396  (class class class)co 6022   0cc0 8925   1c1 8926    < clt 9055   3c3 9984   #chash 11547   USGrph cusg 21234   Cycles ccycl 21382   FriendGrph cfrgra 27743
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-fzo 11068  df-hash 11548  df-word 11652  df-usgra 21236  df-wlk 21383  df-trail 21384  df-pth 21385  df-cycl 21388  df-frgra 27744
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