Theorem 1pthond 27004

Description: In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)

Hypotheses

Ref	Expression
1wlkd.p	|- P = <" X Y ">
1wlkd.f	|- F = <" J ">
1wlkd.x	|- ( ph -> X e. V )
1wlkd.y	|- ( ph -> Y e. V )
1wlkd.l	|- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )
1wlkd.j	|- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )
1wlkd.v	|- V = (Vtx ` G )
1wlkd.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
1pthond	|- ( ph -> F ( X (PathsOn ` G ) Y ) P )

Proof of Theorem 1pthond

Step	Hyp	Ref	Expression
1		1wlkd.p	. . . . 5 |- P = <" X Y ">
2		1wlkd.f	. . . . 5 |- F = <" J ">
3		1wlkd.x	. . . . 5 |- ( ph -> X e. V )
4		1wlkd.y	. . . . 5 |- ( ph -> Y e. V )
5		1wlkd.l	. . . . 5 |- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )
6		1wlkd.j	. . . . 5 |- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )
7		1wlkd.v	. . . . 5 |- V = (Vtx ` G )
8		1wlkd.i	. . . . 5 |- I = (iEdg ` G )
9	1, 2, 3, 4, 5, 6, 7, 8	1wlkd 27001	. . . 4 |- ( ph -> F (Walks ` G ) P )
10	1	fveq1i 6192	. . . . . 6 |- ( P ` 0 ) = ( <" X Y "> ` 0 )
11		s2fv0 13632	. . . . . 6 |- ( X e. V -> ( <" X Y "> ` 0 ) = X )
12	10, 11	syl5eq 2668	. . . . 5 |- ( X e. V -> ( P ` 0 ) = X )
13	3, 12	syl 17	. . . 4 |- ( ph -> ( P ` 0 ) = X )
14	2	fveq2i 6194	. . . . . . 7 |- ( # ` F ) = ( # ` <" J "> )
15		s1len 13385	. . . . . . 7 |- ( # ` <" J "> ) = 1
16	14, 15	eqtri 2644	. . . . . 6 |- ( # ` F ) = 1
17	1, 16	fveq12i 6196	. . . . 5 |- ( P ` ( # ` F ) ) = ( <" X Y "> ` 1 )
18		s2fv1 13633	. . . . . 6 |- ( Y e. V -> ( <" X Y "> ` 1 ) = Y )
19	4, 18	syl 17	. . . . 5 |- ( ph -> ( <" X Y "> ` 1 ) = Y )
20	17, 19	syl5eq 2668	. . . 4 |- ( ph -> ( P ` ( # ` F ) ) = Y )
21		wlkv 26508	. . . . . . 7 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
22		3simpc 1060	. . . . . . 7 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F e. _V /\ P e. _V ) )
23	9, 21, 22	3syl 18	. . . . . 6 |- ( ph -> ( F e. _V /\ P e. _V ) )
24	3, 4, 23	jca31 557	. . . . 5 |- ( ph -> ( ( X e. V /\ Y e. V ) /\ ( F e. _V /\ P e. _V ) ) )
25	7	iswlkon 26553	. . . . 5 |- ( ( ( X e. V /\ Y e. V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( X (WalksOn ` G ) Y ) P <-> ( F (Walks ` G ) P /\ ( P ` 0 ) = X /\ ( P ` ( # ` F ) ) = Y ) ) )
26	24, 25	syl 17	. . . 4 |- ( ph -> ( F ( X (WalksOn ` G ) Y ) P <-> ( F (Walks ` G ) P /\ ( P ` 0 ) = X /\ ( P ` ( # ` F ) ) = Y ) ) )
27	9, 13, 20, 26	mpbir3and 1245	. . 3 |- ( ph -> F ( X (WalksOn ` G ) Y ) P )
28	1, 2, 3, 4, 5, 6, 7, 8	1trld 27002	. . 3 |- ( ph -> F (Trails ` G ) P )
29	7	istrlson 26603	. . . 4 |- ( ( ( X e. V /\ Y e. V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( X (TrailsOn ` G ) Y ) P <-> ( F ( X (WalksOn ` G ) Y ) P /\ F (Trails ` G ) P ) ) )
30	24, 29	syl 17	. . 3 |- ( ph -> ( F ( X (TrailsOn ` G ) Y ) P <-> ( F ( X (WalksOn ` G ) Y ) P /\ F (Trails ` G ) P ) ) )
31	27, 28, 30	mpbir2and 957	. 2 |- ( ph -> F ( X (TrailsOn ` G ) Y ) P )
32	1, 2, 3, 4, 5, 6, 7, 8	1pthd 27003	. 2 |- ( ph -> F (Paths ` G ) P )
33	3	adantl 482	. . . . . . 7 |- ( ( ( F e. _V /\ P e. _V ) /\ ph ) -> X e. V )
34	4	adantl 482	. . . . . . 7 |- ( ( ( F e. _V /\ P e. _V ) /\ ph ) -> Y e. V )
35		simpl 473	. . . . . . 7 |- ( ( ( F e. _V /\ P e. _V ) /\ ph ) -> ( F e. _V /\ P e. _V ) )
36	33, 34, 35	jca31 557	. . . . . 6 |- ( ( ( F e. _V /\ P e. _V ) /\ ph ) -> ( ( X e. V /\ Y e. V ) /\ ( F e. _V /\ P e. _V ) ) )
37	36	ex 450	. . . . 5 |- ( ( F e. _V /\ P e. _V ) -> ( ph -> ( ( X e. V /\ Y e. V ) /\ ( F e. _V /\ P e. _V ) ) ) )
38	21, 22, 37	3syl 18	. . . 4 |- ( F (Walks ` G ) P -> ( ph -> ( ( X e. V /\ Y e. V ) /\ ( F e. _V /\ P e. _V ) ) ) )
39	9, 38	mpcom 38	. . 3 |- ( ph -> ( ( X e. V /\ Y e. V ) /\ ( F e. _V /\ P e. _V ) ) )
40	7	ispthson 26638	. . 3 |- ( ( ( X e. V /\ Y e. V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( X (PathsOn ` G ) Y ) P <-> ( F ( X (TrailsOn ` G ) Y ) P /\ F (Paths ` G ) P ) ) )
41	39, 40	syl 17	. 2 |- ( ph -> ( F ( X (PathsOn ` G ) Y ) P <-> ( F ( X (TrailsOn ` G ) Y ) P /\ F (Paths ` G ) P ) ) )
42	31, 32, 41	mpbir2and 957	1 |- ( ph -> F ( X (PathsOn ` G ) Y ) P )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   #chash 13117   <"cs1 13294   <"cs2 13586  Vtxcvtx 25874  iEdgciedg 25875  Walkscwlks 26492  WalksOncwlkson 26493  Trailsctrls 26587  TrailsOnctrlson 26588  Pathscpths 26608  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-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-pthson 26614

This theorem is referenced by:  upgr1pthond  27010  lppthon  27011  1pthon2v  27013