Theorem 1pthon2v 27013

Description: For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)

Hypotheses

Ref	Expression
1pthon2v.v	|- V = (Vtx ` G )
1pthon2v.e	|- E = (Edg ` G )

Assertion

Ref	Expression
1pthon2v	|- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ E. e e. E { A , B } C_ e ) -> E. f E. p f ( A (PathsOn ` G ) B ) p )

Distinct variable groups:   A, e, f, p   B, e, f, p   e, G, f, p   e, V

Allowed substitution hints:   E( e, f, p)   V( f, p)

Proof of Theorem 1pthon2v

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . . 8 |- ( ( A e. V /\ B e. V ) -> A e. V )
2	1	anim2i 593	. . . . . . 7 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) ) -> ( G e. UHGraph /\ A e. V ) )
3	2	3adant3 1081	. . . . . 6 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ E. e e. E { A , B } C_ e ) -> ( G e. UHGraph /\ A e. V ) )
4	3	adantl 482	. . . . 5 |- ( ( A = B /\ ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ E. e e. E { A , B } C_ e ) ) -> ( G e. UHGraph /\ A e. V ) )
5		1pthon2v.v	. . . . . . 7 |- V = (Vtx ` G )
6	5	0pthonv 26990	. . . . . 6 |- ( A e. V -> E. f E. p f ( A (PathsOn ` G ) A ) p )
7	6	adantl 482	. . . . 5 |- ( ( G e. UHGraph /\ A e. V ) -> E. f E. p f ( A (PathsOn ` G ) A ) p )
8	4, 7	syl 17	. . . 4 |- ( ( A = B /\ ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ E. e e. E { A , B } C_ e ) ) -> E. f E. p f ( A (PathsOn ` G ) A ) p )
9		oveq2 6658	. . . . . . . 8 |- ( B = A -> ( A (PathsOn ` G ) B ) = ( A (PathsOn ` G ) A ) )
10	9	eqcoms 2630	. . . . . . 7 |- ( A = B -> ( A (PathsOn ` G ) B ) = ( A (PathsOn ` G ) A ) )
11	10	breqd 4664	. . . . . 6 |- ( A = B -> ( f ( A (PathsOn ` G ) B ) p <-> f ( A (PathsOn ` G ) A ) p ) )
12	11	2exbidv 1852	. . . . 5 |- ( A = B -> ( E. f E. p f ( A (PathsOn ` G ) B ) p <-> E. f E. p f ( A (PathsOn ` G ) A ) p ) )
13	12	adantr 481	. . . 4 |- ( ( A = B /\ ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ E. e e. E { A , B } C_ e ) ) -> ( E. f E. p f ( A (PathsOn ` G ) B ) p <-> E. f E. p f ( A (PathsOn ` G ) A ) p ) )
14	8, 13	mpbird 247	. . 3 |- ( ( A = B /\ ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ E. e e. E { A , B } C_ e ) ) -> E. f E. p f ( A (PathsOn ` G ) B ) p )
15	14	ex 450	. 2 |- ( A = B -> ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ E. e e. E { A , B } C_ e ) -> E. f E. p f ( A (PathsOn ` G ) B ) p ) )
16		1pthon2v.e	. . . . . . . . . . 11 |- E = (Edg ` G )
17	16	eleq2i 2693	. . . . . . . . . 10 |- ( e e. E <-> e e. (Edg ` G ) )
18		eqid 2622	. . . . . . . . . . 11 |- (iEdg ` G ) = (iEdg ` G )
19	18	uhgredgiedgb 26021	. . . . . . . . . 10 |- ( G e. UHGraph -> ( e e. (Edg ` G ) <-> E. i e. dom (iEdg ` G ) e = ( (iEdg ` G ) ` i ) ) )
20	17, 19	syl5bb 272	. . . . . . . . 9 |- ( G e. UHGraph -> ( e e. E <-> E. i e. dom (iEdg ` G ) e = ( (iEdg ` G ) ` i ) ) )
21	20	3ad2ant1 1082	. . . . . . . 8 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( e e. E <-> E. i e. dom (iEdg ` G ) e = ( (iEdg ` G ) ` i ) ) )
22		s1cli 13384	. . . . . . . . . . . 12 |- <" i "> e. Word _V
23		s2cli 13625	. . . . . . . . . . . 12 |- <" A B "> e. Word _V
24	22, 23	pm3.2i 471	. . . . . . . . . . 11 |- ( <" i "> e. Word _V /\ <" A B "> e. Word _V )
25		eqid 2622	. . . . . . . . . . . 12 |- <" A B "> = <" A B ">
26		eqid 2622	. . . . . . . . . . . 12 |- <" i "> = <" i ">
27		simpl2l 1114	. . . . . . . . . . . 12 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( i e. dom (iEdg ` G ) /\ e = ( (iEdg ` G ) ` i ) ) /\ { A , B } C_ e ) ) -> A e. V )
28		simpl2r 1115	. . . . . . . . . . . 12 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( i e. dom (iEdg ` G ) /\ e = ( (iEdg ` G ) ` i ) ) /\ { A , B } C_ e ) ) -> B e. V )
29		eqneqall 2805	. . . . . . . . . . . . . . . 16 |- ( A = B -> ( A =/= B -> ( (iEdg ` G ) ` i ) = { A } ) )
30	29	com12 32	. . . . . . . . . . . . . . 15 |- ( A =/= B -> ( A = B -> ( (iEdg ` G ) ` i ) = { A } ) )
31	30	3ad2ant3 1084	. . . . . . . . . . . . . 14 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( A = B -> ( (iEdg ` G ) ` i ) = { A } ) )
32	31	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( i e. dom (iEdg ` G ) /\ e = ( (iEdg ` G ) ` i ) ) /\ { A , B } C_ e ) ) -> ( A = B -> ( (iEdg ` G ) ` i ) = { A } ) )
33	32	imp 445	. . . . . . . . . . . 12 |- ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( i e. dom (iEdg ` G ) /\ e = ( (iEdg ` G ) ` i ) ) /\ { A , B } C_ e ) ) /\ A = B ) -> ( (iEdg ` G ) ` i ) = { A } )
34		sseq2 3627	. . . . . . . . . . . . . . . 16 |- ( e = ( (iEdg ` G ) ` i ) -> ( { A , B } C_ e <-> { A , B } C_ ( (iEdg ` G ) ` i ) ) )
35	34	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( i e. dom (iEdg ` G ) /\ e = ( (iEdg ` G ) ` i ) ) -> ( { A , B } C_ e <-> { A , B } C_ ( (iEdg ` G ) ` i ) ) )
36	35	biimpa 501	. . . . . . . . . . . . . 14 |- ( ( ( i e. dom (iEdg ` G ) /\ e = ( (iEdg ` G ) ` i ) ) /\ { A , B } C_ e ) -> { A , B } C_ ( (iEdg ` G ) ` i ) )
37	36	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( i e. dom (iEdg ` G ) /\ e = ( (iEdg ` G ) ` i ) ) /\ { A , B } C_ e ) ) -> { A , B } C_ ( (iEdg ` G ) ` i ) )
38	37	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( i e. dom (iEdg ` G ) /\ e = ( (iEdg ` G ) ` i ) ) /\ { A , B } C_ e ) ) /\ A =/= B ) -> { A , B } C_ ( (iEdg ` G ) ` i ) )
39	25, 26, 27, 28, 33, 38, 5, 18	1pthond 27004	. . . . . . . . . . 11 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( i e. dom (iEdg ` G ) /\ e = ( (iEdg ` G ) ` i ) ) /\ { A , B } C_ e ) ) -> <" i "> ( A (PathsOn ` G ) B ) <" A B "> )
40		breq12 4658	. . . . . . . . . . . 12 |- ( ( f = <" i "> /\ p = <" A B "> ) -> ( f ( A (PathsOn ` G ) B ) p <-> <" i "> ( A (PathsOn ` G ) B ) <" A B "> ) )
41	40	spc2egv 3295	. . . . . . . . . . 11 |- ( ( <" i "> e. Word _V /\ <" A B "> e. Word _V ) -> ( <" i "> ( A (PathsOn ` G ) B ) <" A B "> -> E. f E. p f ( A (PathsOn ` G ) B ) p ) )
42	24, 39, 41	mpsyl 68	. . . . . . . . . 10 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( i e. dom (iEdg ` G ) /\ e = ( (iEdg ` G ) ` i ) ) /\ { A , B } C_ e ) ) -> E. f E. p f ( A (PathsOn ` G ) B ) p )
43	42	exp44 641	. . . . . . . . 9 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( i e. dom (iEdg ` G ) -> ( e = ( (iEdg ` G ) ` i ) -> ( { A , B } C_ e -> E. f E. p f ( A (PathsOn ` G ) B ) p ) ) ) )
44	43	rexlimdv 3030	. . . . . . . 8 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( E. i e. dom (iEdg ` G ) e = ( (iEdg ` G ) ` i ) -> ( { A , B } C_ e -> E. f E. p f ( A (PathsOn ` G ) B ) p ) ) )
45	21, 44	sylbid 230	. . . . . . 7 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( e e. E -> ( { A , B } C_ e -> E. f E. p f ( A (PathsOn ` G ) B ) p ) ) )
46	45	rexlimdv 3030	. . . . . 6 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( E. e e. E { A , B } C_ e -> E. f E. p f ( A (PathsOn ` G ) B ) p ) )
47	46	3exp 1264	. . . . 5 |- ( G e. UHGraph -> ( ( A e. V /\ B e. V ) -> ( A =/= B -> ( E. e e. E { A , B } C_ e -> E. f E. p f ( A (PathsOn ` G ) B ) p ) ) ) )
48	47	com34 91	. . . 4 |- ( G e. UHGraph -> ( ( A e. V /\ B e. V ) -> ( E. e e. E { A , B } C_ e -> ( A =/= B -> E. f E. p f ( A (PathsOn ` G ) B ) p ) ) ) )
49	48	3imp 1256	. . 3 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ E. e e. E { A , B } C_ e ) -> ( A =/= B -> E. f E. p f ( A (PathsOn ` G ) B ) p ) )
50	49	com12 32	. 2 |- ( A =/= B -> ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ E. e e. E { A , B } C_ e ) -> E. f E. p f ( A (PathsOn ` G ) B ) p ) )
51	15, 50	pm2.61ine 2877	1 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ E. e e. E { A , B } C_ e ) -> E. f E. p f ( A (PathsOn ` G ) B ) p )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650  Word cword 13291   <"cs1 13294   <"cs2 13586  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951  PathsOncpthson 26610

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-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-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-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-edg 25940  df-uhgr 25953  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-pthson 26614

This theorem is referenced by:  1pthon2ve  27014  cusconngr  27051