Theorem wlkonl1iedg 26561

Description: If there is a walk between two vertices A and B at least of length 1, then the start vertex A is incident with an edge. (Contributed by AV, 4-Apr-2021.)

Hypothesis

Ref	Expression
wlkonl1iedg.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
wlkonl1iedg	|- ( ( F ( A (WalksOn ` G ) B ) P /\ ( # ` F ) =/= 0 ) -> E. e e. ran I A e. e )

Distinct variable groups:   A, e   e, F   e, G   e, I   P, e

Allowed substitution hint:   B( e)

Proof of Theorem wlkonl1iedg

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
2	1	wlkonprop 26554	. . 3 |- ( F ( A (WalksOn ` G ) B ) P -> ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) )
3		fveq2 6191	. . . . . . . . . . 11 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
4		oveq1 6657	. . . . . . . . . . . . 13 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
5		0p1e1 11132	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
6	4, 5	syl6eq 2672	. . . . . . . . . . . 12 |- ( k = 0 -> ( k + 1 ) = 1 )
7	6	fveq2d 6195	. . . . . . . . . . 11 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
8	3, 7	preq12d 4276	. . . . . . . . . 10 |- ( k = 0 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
9	8	sseq1d 3632	. . . . . . . . 9 |- ( k = 0 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e <-> { ( P ` 0 ) , ( P ` 1 ) } C_ e ) )
10	9	rexbidv 3052	. . . . . . . 8 |- ( k = 0 -> ( E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e <-> E. e e. ran I { ( P ` 0 ) , ( P ` 1 ) } C_ e ) )
11		wlkonl1iedg.i	. . . . . . . . . . 11 |- I = (iEdg ` G )
12	11	wlkvtxiedg 26520	. . . . . . . . . 10 |- ( F (Walks ` G ) P -> A. k e. ( 0..^ ( # ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )
13	12	adantr 481	. . . . . . . . 9 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) -> A. k e. ( 0..^ ( # ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )
14	13	adantr 481	. . . . . . . 8 |- ( ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) /\ ( # ` F ) =/= 0 ) -> A. k e. ( 0..^ ( # ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )
15		wlkcl 26511	. . . . . . . . . . 11 |- ( F (Walks ` G ) P -> ( # ` F ) e. NN0 )
16		elnnne0 11306	. . . . . . . . . . . . 13 |- ( ( # ` F ) e. NN <-> ( ( # ` F ) e. NN0 /\ ( # ` F ) =/= 0 ) )
17	16	simplbi2 655	. . . . . . . . . . . 12 |- ( ( # ` F ) e. NN0 -> ( ( # ` F ) =/= 0 -> ( # ` F ) e. NN ) )
18		lbfzo0 12507	. . . . . . . . . . . 12 |- ( 0 e. ( 0..^ ( # ` F ) ) <-> ( # ` F ) e. NN )
19	17, 18	syl6ibr 242	. . . . . . . . . . 11 |- ( ( # ` F ) e. NN0 -> ( ( # ` F ) =/= 0 -> 0 e. ( 0..^ ( # ` F ) ) ) )
20	15, 19	syl 17	. . . . . . . . . 10 |- ( F (Walks ` G ) P -> ( ( # ` F ) =/= 0 -> 0 e. ( 0..^ ( # ` F ) ) ) )
21	20	adantr 481	. . . . . . . . 9 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) -> ( ( # ` F ) =/= 0 -> 0 e. ( 0..^ ( # ` F ) ) ) )
22	21	imp 445	. . . . . . . 8 |- ( ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) /\ ( # ` F ) =/= 0 ) -> 0 e. ( 0..^ ( # ` F ) ) )
23	10, 14, 22	rspcdva 3316	. . . . . . 7 |- ( ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) /\ ( # ` F ) =/= 0 ) -> E. e e. ran I { ( P ` 0 ) , ( P ` 1 ) } C_ e )
24		fvex 6201	. . . . . . . . . . 11 |- ( P ` 0 ) e. _V
25		fvex 6201	. . . . . . . . . . 11 |- ( P ` 1 ) e. _V
26	24, 25	prss 4351	. . . . . . . . . 10 |- ( ( ( P ` 0 ) e. e /\ ( P ` 1 ) e. e ) <-> { ( P ` 0 ) , ( P ` 1 ) } C_ e )
27		eleq1 2689	. . . . . . . . . . . . 13 |- ( ( P ` 0 ) = A -> ( ( P ` 0 ) e. e <-> A e. e ) )
28		ax-1 6	. . . . . . . . . . . . 13 |- ( A e. e -> ( ( P ` 1 ) e. e -> A e. e ) )
29	27, 28	syl6bi 243	. . . . . . . . . . . 12 |- ( ( P ` 0 ) = A -> ( ( P ` 0 ) e. e -> ( ( P ` 1 ) e. e -> A e. e ) ) )
30	29	adantl 482	. . . . . . . . . . 11 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) -> ( ( P ` 0 ) e. e -> ( ( P ` 1 ) e. e -> A e. e ) ) )
31	30	impd 447	. . . . . . . . . 10 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) -> ( ( ( P ` 0 ) e. e /\ ( P ` 1 ) e. e ) -> A e. e ) )
32	26, 31	syl5bir 233	. . . . . . . . 9 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) -> ( { ( P ` 0 ) , ( P ` 1 ) } C_ e -> A e. e ) )
33	32	reximdv 3016	. . . . . . . 8 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) -> ( E. e e. ran I { ( P ` 0 ) , ( P ` 1 ) } C_ e -> E. e e. ran I A e. e ) )
34	33	adantr 481	. . . . . . 7 |- ( ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) /\ ( # ` F ) =/= 0 ) -> ( E. e e. ran I { ( P ` 0 ) , ( P ` 1 ) } C_ e -> E. e e. ran I A e. e ) )
35	23, 34	mpd 15	. . . . . 6 |- ( ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) /\ ( # ` F ) =/= 0 ) -> E. e e. ran I A e. e )
36	35	ex 450	. . . . 5 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A ) -> ( ( # ` F ) =/= 0 -> E. e e. ran I A e. e ) )
37	36	3adant3 1081	. . . 4 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( ( # ` F ) =/= 0 -> E. e e. ran I A e. e ) )
38	37	3ad2ant3 1084	. . 3 |- ( ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( ( # ` F ) =/= 0 -> E. e e. ran I A e. e ) )
39	2, 38	syl 17	. 2 |- ( F ( A (WalksOn ` G ) B ) P -> ( ( # ` F ) =/= 0 -> E. e e. ran I A e. e ) )
40	39	imp 445	1 |- ( ( F ( A (WalksOn ` G ) B ) P /\ ( # ` F ) =/= 0 ) -> E. e e. ran I A e. e )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {cpr 4179   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NNcn 11020   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Walkscwlks 26492  WalksOncwlkson 26493

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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495  df-wlkson 26496

This theorem is referenced by:  conngrv2edg  27055