Theorem wpthswwlks2on 26854

Description: For two different vertices, a walk of length 2 between these vertices is a simple path of length 2 between these vertices in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 13-May-2021.)

Hypothesis

Ref	Expression
wpthswwlks2on.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
wpthswwlks2on	|- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( A ( 2 WSPathsNOn G ) B ) = ( A ( 2 WWalksNOn G ) B ) )

Proof of Theorem wpthswwlks2on

Dummy variables f w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wpthswwlks2on.v	. . . . . . . 8 |- V = (Vtx ` G )
2	1	wwlknon 26742	. . . . . . 7 |- ( ( A e. V /\ B e. V ) -> ( w e. ( A ( 2 WWalksNOn G ) B ) <-> ( w e. ( 2 WWalksN G ) /\ ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) )
3	2	3ad2ant2 1083	. . . . . 6 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( w e. ( A ( 2 WWalksNOn G ) B ) <-> ( w e. ( 2 WWalksN G ) /\ ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) )
4	3	anbi1d 741	. . . . 5 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( ( w e. ( A ( 2 WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) <-> ( ( w e. ( 2 WWalksN G ) /\ ( w ` 0 ) = A /\ ( w ` 2 ) = B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) ) )
5		3anass 1042	. . . . . . 7 |- ( ( w e. ( 2 WWalksN G ) /\ ( w ` 0 ) = A /\ ( w ` 2 ) = B ) <-> ( w e. ( 2 WWalksN G ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) )
6	5	anbi1i 731	. . . . . 6 |- ( ( ( w e. ( 2 WWalksN G ) /\ ( w ` 0 ) = A /\ ( w ` 2 ) = B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) <-> ( ( w e. ( 2 WWalksN G ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) )
7		anass 681	. . . . . 6 |- ( ( ( w e. ( 2 WWalksN G ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) <-> ( w e. ( 2 WWalksN G ) /\ ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) ) )
8	6, 7	bitri 264	. . . . 5 |- ( ( ( w e. ( 2 WWalksN G ) /\ ( w ` 0 ) = A /\ ( w ` 2 ) = B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) <-> ( w e. ( 2 WWalksN G ) /\ ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) ) )
9	4, 8	syl6bb 276	. . . 4 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( ( w e. ( A ( 2 WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) <-> ( w e. ( 2 WWalksN G ) /\ ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) ) ) )
10	9	rabbidva2 3186	. . 3 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> { w e. ( A ( 2 WWalksNOn G ) B ) | E. f f ( A (SPathsOn ` G ) B ) w } = { w e. ( 2 WWalksN G ) | ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) } )
11		usgrupgr 26077	. . . . . . . . . . 11 |- ( G e. USGraph -> G e. UPGraph )
12		wlklnwwlknupgr 26772	. . . . . . . . . . 11 |- ( G e. UPGraph -> ( E. f ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) <-> w e. ( 2 WWalksN G ) ) )
13	11, 12	syl 17	. . . . . . . . . 10 |- ( G e. USGraph -> ( E. f ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) <-> w e. ( 2 WWalksN G ) ) )
14	13	bicomd 213	. . . . . . . . 9 |- ( G e. USGraph -> ( w e. ( 2 WWalksN G ) <-> E. f ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) )
15	14	3ad2ant1 1082	. . . . . . . 8 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( w e. ( 2 WWalksN G ) <-> E. f ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) )
16		simprl 794	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> f (Walks ` G ) w )
17		simprl 794	. . . . . . . . . . . . . . 15 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) -> ( w ` 0 ) = A )
18	17	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> ( w ` 0 ) = A )
19		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( ( # ` f ) = 2 -> ( w ` ( # ` f ) ) = ( w ` 2 ) )
20	19	ad2antll 765	. . . . . . . . . . . . . . 15 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> ( w ` ( # ` f ) ) = ( w ` 2 ) )
21		simprr 796	. . . . . . . . . . . . . . . 16 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) -> ( w ` 2 ) = B )
22	21	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> ( w ` 2 ) = B )
23	20, 22	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> ( w ` ( # ` f ) ) = B )
24		simpll2 1101	. . . . . . . . . . . . . . 15 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> ( A e. V /\ B e. V ) )
25		vex 3203	. . . . . . . . . . . . . . . 16 |- f e. _V
26		vex 3203	. . . . . . . . . . . . . . . 16 |- w e. _V
27	25, 26	pm3.2i 471	. . . . . . . . . . . . . . 15 |- ( f e. _V /\ w e. _V )
28	1	iswlkon 26553	. . . . . . . . . . . . . . 15 |- ( ( ( A e. V /\ B e. V ) /\ ( f e. _V /\ w e. _V ) ) -> ( f ( A (WalksOn ` G ) B ) w <-> ( f (Walks ` G ) w /\ ( w ` 0 ) = A /\ ( w ` ( # ` f ) ) = B ) ) )
29	24, 27, 28	sylancl 694	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> ( f ( A (WalksOn ` G ) B ) w <-> ( f (Walks ` G ) w /\ ( w ` 0 ) = A /\ ( w ` ( # ` f ) ) = B ) ) )
30	16, 18, 23, 29	mpbir3and 1245	. . . . . . . . . . . . 13 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> f ( A (WalksOn ` G ) B ) w )
31		simpll1 1100	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> G e. USGraph )
32		simprr 796	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> ( # ` f ) = 2 )
33		simpll3 1102	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> A =/= B )
34		usgr2wlkspth 26655	. . . . . . . . . . . . . 14 |- ( ( G e. USGraph /\ ( # ` f ) = 2 /\ A =/= B ) -> ( f ( A (WalksOn ` G ) B ) w <-> f ( A (SPathsOn ` G ) B ) w ) )
35	31, 32, 33, 34	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> ( f ( A (WalksOn ` G ) B ) w <-> f ( A (SPathsOn ` G ) B ) w ) )
36	30, 35	mpbid 222	. . . . . . . . . . . 12 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) /\ ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) ) -> f ( A (SPathsOn ` G ) B ) w )
37	36	ex 450	. . . . . . . . . . 11 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) -> ( ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) -> f ( A (SPathsOn ` G ) B ) w ) )
38	37	eximdv 1846	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) -> ( E. f ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) -> E. f f ( A (SPathsOn ` G ) B ) w ) )
39	38	ex 450	. . . . . . . . 9 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) -> ( E. f ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) -> E. f f ( A (SPathsOn ` G ) B ) w ) ) )
40	39	com23 86	. . . . . . . 8 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( E. f ( f (Walks ` G ) w /\ ( # ` f ) = 2 ) -> ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) -> E. f f ( A (SPathsOn ` G ) B ) w ) ) )
41	15, 40	sylbid 230	. . . . . . 7 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( w e. ( 2 WWalksN G ) -> ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) -> E. f f ( A (SPathsOn ` G ) B ) w ) ) )
42	41	imp 445	. . . . . 6 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ w e. ( 2 WWalksN G ) ) -> ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) -> E. f f ( A (SPathsOn ` G ) B ) w ) )
43	42	pm4.71d 666	. . . . 5 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ w e. ( 2 WWalksN G ) ) -> ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) <-> ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) ) )
44	43	bicomd 213	. . . 4 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) /\ w e. ( 2 WWalksN G ) ) -> ( ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) <-> ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) ) )
45	44	rabbidva 3188	. . 3 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> { w e. ( 2 WWalksN G ) | ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) /\ E. f f ( A (SPathsOn ` G ) B ) w ) } = { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) } )
46	10, 45	eqtrd 2656	. 2 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> { w e. ( A ( 2 WWalksNOn G ) B ) | E. f f ( A (SPathsOn ` G ) B ) w } = { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) } )
47	1	iswspthsnon 26741	. . 3 |- ( ( A e. V /\ B e. V ) -> ( A ( 2 WSPathsNOn G ) B ) = { w e. ( A ( 2 WWalksNOn G ) B ) | E. f f ( A (SPathsOn ` G ) B ) w } )
48	47	3ad2ant2 1083	. 2 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( A ( 2 WSPathsNOn G ) B ) = { w e. ( A ( 2 WWalksNOn G ) B ) | E. f f ( A (SPathsOn ` G ) B ) w } )
49	1	iswwlksnon 26740	. . 3 |- ( ( A e. V /\ B e. V ) -> ( A ( 2 WWalksNOn G ) B ) = { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) } )
50	49	3ad2ant2 1083	. 2 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( A ( 2 WWalksNOn G ) B ) = { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = B ) } )
51	46, 48, 50	3eqtr4d 2666	1 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( A ( 2 WSPathsNOn G ) B ) = ( A ( 2 WWalksNOn G ) B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   2c2 11070   #chash 13117  Vtxcvtx 25874   UPGraph cupgr 25975   USGraph cusgr 26044  Walkscwlks 26492  WalksOncwlkson 26493  SPathsOncspthson 26611   WWalksN cwwlksn 26718   WWalksNOn cwwlksnon 26719   WSPathsNOn cwwspthsnon 26721

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-ac2 9285  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-se 5074  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-isom 5897  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-2o 7561  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-ac 8939  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-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-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-pthson 26614  df-spthson 26615  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-wspthsnon 26726

This theorem is referenced by:  usgr2wspthons3  26857  frgr2wsp1  27194