Theorem 3cyclfrgr 27152

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.) (Revised by AV, 2-Apr-2021.)

Hypothesis

Ref	Expression
3cyclfrgr.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
3cyclfrgr	|- ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) -> A. v e. V E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = v ) )

Distinct variable groups:   f, G, p, v   v, V

Allowed substitution hints:   V( f, p)

Proof of Theorem 3cyclfrgr

Dummy variables b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3cyclfrgr.v	. . 3 |- V = (Vtx ` G )
2		eqid 2622	. . 3 |- (Edg ` G ) = (Edg ` G )
3	1, 2	3cyclfrgrrn 27150	. 2 |- ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) -> A. v e. V E. b e. V E. c e. V ( { v , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , v } e. (Edg ` G ) ) )
4		frgrusgr 27124	. . . . . . . 8 |- ( G e. FriendGraph -> G e. USGraph )
5		usgrumgr 26074	. . . . . . . 8 |- ( G e. USGraph -> G e. UMGraph )
6	4, 5	syl 17	. . . . . . 7 |- ( G e. FriendGraph -> G e. UMGraph )
7	6	ad4antr 768	. . . . . 6 |- ( ( ( ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) /\ v e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( { v , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , v } e. (Edg ` G ) ) ) -> G e. UMGraph )
8		simpr 477	. . . . . . . . 9 |- ( ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) /\ v e. V ) -> v e. V )
9	8	anim1i 592	. . . . . . . 8 |- ( ( ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) /\ v e. V ) /\ ( b e. V /\ c e. V ) ) -> ( v e. V /\ ( b e. V /\ c e. V ) ) )
10		3anass 1042	. . . . . . . 8 |- ( ( v e. V /\ b e. V /\ c e. V ) <-> ( v e. V /\ ( b e. V /\ c e. V ) ) )
11	9, 10	sylibr 224	. . . . . . 7 |- ( ( ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) /\ v e. V ) /\ ( b e. V /\ c e. V ) ) -> ( v e. V /\ b e. V /\ c e. V ) )
12	11	adantr 481	. . . . . 6 |- ( ( ( ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) /\ v e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( { v , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , v } e. (Edg ` G ) ) ) -> ( v e. V /\ b e. V /\ c e. V ) )
13		simpr 477	. . . . . 6 |- ( ( ( ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) /\ v e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( { v , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , v } e. (Edg ` G ) ) ) -> ( { v , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , v } e. (Edg ` G ) ) )
14	1, 2	umgr3cyclex 27043	. . . . . 6 |- ( ( G e. UMGraph /\ ( v e. V /\ b e. V /\ c e. V ) /\ ( { v , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , v } e. (Edg ` G ) ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = v ) )
15	7, 12, 13, 14	syl3anc 1326	. . . . 5 |- ( ( ( ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) /\ v e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( { v , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , v } e. (Edg ` G ) ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = v ) )
16	15	ex 450	. . . 4 |- ( ( ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) /\ v e. V ) /\ ( b e. V /\ c e. V ) ) -> ( ( { v , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , v } e. (Edg ` G ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = v ) ) )
17	16	rexlimdvva 3038	. . 3 |- ( ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) /\ v e. V ) -> ( E. b e. V E. c e. V ( { v , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , v } e. (Edg ` G ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = v ) ) )
18	17	ralimdva 2962	. 2 |- ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) -> ( A. v e. V E. b e. V E. c e. V ( { v , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , v } e. (Edg ` G ) ) -> A. v e. V E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = v ) ) )
19	3, 18	mpd 15	1 |- ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) -> A. v e. V E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = v ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {cpr 4179   class class class wbr 4653   ` cfv 5888   0cc0 9936   1c1 9937   < clt 10074   3c3 11071   #chash 13117  Vtxcvtx 25874  Edgcedg 25939   UMGraph cumgr 25976   USGraph cusgr 26044  Cyclesccycls 26680   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-s4 13595  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-usgr 26046  df-wlks 26495  df-trls 26589  df-pths 26612  df-cycls 26682  df-frgr 27121

This theorem is referenced by: (None)