Theorem 2pthfrgr 27148

Description: Between any two (different) vertices in a friendship graph, tere is a 2-path (simple path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 1-Apr-2021.)

Hypothesis

Ref	Expression
2pthfrgr.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
2pthfrgr	|- ( G e. FriendGraph -> A. a e. V A. b e. ( V \ { a } ) E. f E. p ( f ( a (SPathsOn ` G ) b ) p /\ ( # ` f ) = 2 ) )

Distinct variable groups:   G, a, b, f, p   V, a, b

Allowed substitution hints:   V( f, p)

Proof of Theorem 2pthfrgr

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2pthfrgr.v	. . 3 |- V = (Vtx ` G )
2		eqid 2622	. . 3 |- (Edg ` G ) = (Edg ` G )
3	1, 2	2pthfrgrrn2 27147	. 2 |- ( G e. FriendGraph -> A. a e. V A. b e. ( V \ { a } ) E. m e. V ( ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) /\ ( a =/= m /\ m =/= b ) ) )
4		frgrusgr 27124	. . . . . . . . . . . . 13 |- ( G e. FriendGraph -> G e. USGraph )
5		usgruhgr 26078	. . . . . . . . . . . . 13 |- ( G e. USGraph -> G e. UHGraph )
6	4, 5	syl 17	. . . . . . . . . . . 12 |- ( G e. FriendGraph -> G e. UHGraph )
7	6	adantr 481	. . . . . . . . . . 11 |- ( ( G e. FriendGraph /\ a e. V ) -> G e. UHGraph )
8	7	adantr 481	. . . . . . . . . 10 |- ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) -> G e. UHGraph )
9	8	adantr 481	. . . . . . . . 9 |- ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) -> G e. UHGraph )
10		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) -> a e. V )
11		simpr 477	. . . . . . . . . 10 |- ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) -> m e. V )
12		eldifi 3732	. . . . . . . . . . 11 |- ( b e. ( V \ { a } ) -> b e. V )
13	12	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) -> b e. V )
14	10, 11, 13	3jca 1242	. . . . . . . . 9 |- ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) -> ( a e. V /\ m e. V /\ b e. V ) )
15	9, 14	jca 554	. . . . . . . 8 |- ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) -> ( G e. UHGraph /\ ( a e. V /\ m e. V /\ b e. V ) ) )
16	15	adantr 481	. . . . . . 7 |- ( ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) /\ ( ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) /\ ( a =/= m /\ m =/= b ) ) ) -> ( G e. UHGraph /\ ( a e. V /\ m e. V /\ b e. V ) ) )
17		simprrl 804	. . . . . . 7 |- ( ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) /\ ( ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) /\ ( a =/= m /\ m =/= b ) ) ) -> a =/= m )
18		eldifsn 4317	. . . . . . . . 9 |- ( b e. ( V \ { a } ) <-> ( b e. V /\ b =/= a ) )
19		necom 2847	. . . . . . . . . 10 |- ( b =/= a <-> a =/= b )
20	19	biimpi 206	. . . . . . . . 9 |- ( b =/= a -> a =/= b )
21	18, 20	simplbiim 659	. . . . . . . 8 |- ( b e. ( V \ { a } ) -> a =/= b )
22	21	ad3antlr 767	. . . . . . 7 |- ( ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) /\ ( ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) /\ ( a =/= m /\ m =/= b ) ) ) -> a =/= b )
23		simprrr 805	. . . . . . 7 |- ( ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) /\ ( ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) /\ ( a =/= m /\ m =/= b ) ) ) -> m =/= b )
24		simprl 794	. . . . . . 7 |- ( ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) /\ ( ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) /\ ( a =/= m /\ m =/= b ) ) ) -> ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) )
25	1, 2	2pthon3v 26839	. . . . . . 7 |- ( ( ( G e. UHGraph /\ ( a e. V /\ m e. V /\ b e. V ) ) /\ ( a =/= m /\ a =/= b /\ m =/= b ) /\ ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) ) -> E. f E. p ( f ( a (SPathsOn ` G ) b ) p /\ ( # ` f ) = 2 ) )
26	16, 17, 22, 23, 24, 25	syl131anc 1339	. . . . . 6 |- ( ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) /\ ( ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) /\ ( a =/= m /\ m =/= b ) ) ) -> E. f E. p ( f ( a (SPathsOn ` G ) b ) p /\ ( # ` f ) = 2 ) )
27	26	ex 450	. . . . 5 |- ( ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) /\ m e. V ) -> ( ( ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) /\ ( a =/= m /\ m =/= b ) ) -> E. f E. p ( f ( a (SPathsOn ` G ) b ) p /\ ( # ` f ) = 2 ) ) )
28	27	rexlimdva 3031	. . . 4 |- ( ( ( G e. FriendGraph /\ a e. V ) /\ b e. ( V \ { a } ) ) -> ( E. m e. V ( ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) /\ ( a =/= m /\ m =/= b ) ) -> E. f E. p ( f ( a (SPathsOn ` G ) b ) p /\ ( # ` f ) = 2 ) ) )
29	28	ralimdva 2962	. . 3 |- ( ( G e. FriendGraph /\ a e. V ) -> ( A. b e. ( V \ { a } ) E. m e. V ( ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) /\ ( a =/= m /\ m =/= b ) ) -> A. b e. ( V \ { a } ) E. f E. p ( f ( a (SPathsOn ` G ) b ) p /\ ( # ` f ) = 2 ) ) )
30	29	ralimdva 2962	. 2 |- ( G e. FriendGraph -> ( A. a e. V A. b e. ( V \ { a } ) E. m e. V ( ( { a , m } e. (Edg ` G ) /\ { m , b } e. (Edg ` G ) ) /\ ( a =/= m /\ m =/= b ) ) -> A. a e. V A. b e. ( V \ { a } ) E. f E. p ( f ( a (SPathsOn ` G ) b ) p /\ ( # ` f ) = 2 ) ) )
31	3, 30	mpd 15	1 |- ( G e. FriendGraph -> A. a e. V A. b e. ( V \ { a } ) E. f E. p ( f ( a (SPathsOn ` G ) b ) p /\ ( # ` f ) = 2 ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   {csn 4177   {cpr 4179   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   2c2 11070   #chash 13117  Vtxcvtx 25874  Edgcedg 25939   UHGraph cuhgr 25951   USGraph cusgr 26044  SPathsOncspthson 26611   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-n0 11293  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-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-spthson 26615  df-frgr 27121

This theorem is referenced by:  frgrconngr  27158