Theorem wwlksnextbij 26797

Description: There is a bijection between the extensions of a walk (as word) by an edge and the set of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 18-Apr-2021.)

Hypotheses

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

Assertion

Ref	Expression
wwlksnextbij	|- ( W e. ( N WWalksN G ) -> E. f f : { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. E } )

Distinct variable groups:   f, E, n, w   f, G, w   f, N, w   f, V, n, w   f, W, n, w

Allowed substitution hints:   G( n)   N( n)

Proof of Theorem wwlksnextbij

Dummy variables p t x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 6680	. . 3 |- ( W e. ( N WWalksN G ) -> ( ( N + 1 ) WWalksN G ) e. _V )
2		rabexg 4812	. . 3 |- ( ( ( N + 1 ) WWalksN G ) e. _V -> { t e. ( ( N + 1 ) WWalksN G ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } e. _V )
3		mptexg 6484	. . 3 |- ( { t e. ( ( N + 1 ) WWalksN G ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } e. _V -> ( x e. { t e. ( ( N + 1 ) WWalksN G ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } |-> ( lastS ` x ) ) e. _V )
4	1, 2, 3	3syl 18	. 2 |- ( W e. ( N WWalksN G ) -> ( x e. { t e. ( ( N + 1 ) WWalksN G ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } |-> ( lastS ` x ) ) e. _V )
5		wwlksnextbij.v	. . . 4 |- V = (Vtx ` G )
6		wwlksnextbij.e	. . . 4 |- E = (Edg ` G )
7		eqid 2622	. . . 4 |- { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) }
8		preq2 4269	. . . . . 6 |- ( n = p -> { ( lastS ` W ) , n } = { ( lastS ` W ) , p } )
9	8	eleq1d 2686	. . . . 5 |- ( n = p -> ( { ( lastS ` W ) , n } e. E <-> { ( lastS ` W ) , p } e. E ) )
10	9	cbvrabv 3199	. . . 4 |- { n e. V | { ( lastS ` W ) , n } e. E } = { p e. V | { ( lastS ` W ) , p } e. E }
11		fveq2 6191	. . . . . . . 8 |- ( t = w -> ( # ` t ) = ( # ` w ) )
12	11	eqeq1d 2624	. . . . . . 7 |- ( t = w -> ( ( # ` t ) = ( N + 2 ) <-> ( # ` w ) = ( N + 2 ) ) )
13		oveq1 6657	. . . . . . . 8 |- ( t = w -> ( t substr <. 0 , ( N + 1 ) >. ) = ( w substr <. 0 , ( N + 1 ) >. ) )
14	13	eqeq1d 2624	. . . . . . 7 |- ( t = w -> ( ( t substr <. 0 , ( N + 1 ) >. ) = W <-> ( w substr <. 0 , ( N + 1 ) >. ) = W ) )
15		fveq2 6191	. . . . . . . . 9 |- ( t = w -> ( lastS ` t ) = ( lastS ` w ) )
16	15	preq2d 4275	. . . . . . . 8 |- ( t = w -> { ( lastS ` W ) , ( lastS ` t ) } = { ( lastS ` W ) , ( lastS ` w ) } )
17	16	eleq1d 2686	. . . . . . 7 |- ( t = w -> ( { ( lastS ` W ) , ( lastS ` t ) } e. E <-> { ( lastS ` W ) , ( lastS ` w ) } e. E ) )
18	12, 14, 17	3anbi123d 1399	. . . . . 6 |- ( t = w -> ( ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) <-> ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) )
19	18	cbvrabv 3199	. . . . 5 |- { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) }
20	19	mpteq1i 4739	. . . 4 |- ( x e. { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } |-> ( lastS ` x ) ) = ( x e. { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } |-> ( lastS ` x ) )
21	5, 6, 7, 10, 20	wwlksnextbij0 26796	. . 3 |- ( W e. ( N WWalksN G ) -> ( x e. { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } |-> ( lastS ` x ) ) : { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. E } )
22		eqid 2622	. . . . . . 7 |- { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } = { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) }
23	5, 6, 22	wwlksnextwrd 26792	. . . . . 6 |- ( W e. ( N WWalksN G ) -> { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } = { t e. ( ( N + 1 ) WWalksN G ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } )
24	23	eqcomd 2628	. . . . 5 |- ( W e. ( N WWalksN G ) -> { t e. ( ( N + 1 ) WWalksN G ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } = { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } )
25	24	mpteq1d 4738	. . . 4 |- ( W e. ( N WWalksN G ) -> ( x e. { t e. ( ( N + 1 ) WWalksN G ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } |-> ( lastS ` x ) ) = ( x e. { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } |-> ( lastS ` x ) ) )
26	5, 6, 7	wwlksnextwrd 26792	. . . . 5 |- ( W e. ( N WWalksN G ) -> { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } = { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } )
27	26	eqcomd 2628	. . . 4 |- ( W e. ( N WWalksN G ) -> { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } )
28		eqidd 2623	. . . 4 |- ( W e. ( N WWalksN G ) -> { n e. V | { ( lastS ` W ) , n } e. E } = { n e. V | { ( lastS ` W ) , n } e. E } )
29	25, 27, 28	f1oeq123d 6133	. . 3 |- ( W e. ( N WWalksN G ) -> ( ( x e. { t e. ( ( N + 1 ) WWalksN G ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } |-> ( lastS ` x ) ) : { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. E } <-> ( x e. { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } |-> ( lastS ` x ) ) : { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. E } ) )
30	21, 29	mpbird 247	. 2 |- ( W e. ( N WWalksN G ) -> ( x e. { t e. ( ( N + 1 ) WWalksN G ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } |-> ( lastS ` x ) ) : { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. E } )
31		f1oeq1 6127	. 2 |- ( f = ( x e. { t e. ( ( N + 1 ) WWalksN G ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } |-> ( lastS ` x ) ) -> ( f : { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. E } <-> ( x e. { t e. ( ( N + 1 ) WWalksN G ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. E ) } |-> ( lastS ` x ) ) : { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. E } ) )
32	4, 30, 31	elabd 3352	1 |- ( W e. ( N WWalksN G ) -> E. f f : { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. E } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {crab 2916   _Vcvv 3200   {cpr 4179   <.cop 4183   |-> cmpt 4729   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   #chash 13117  Word cword 13291   lastS clsw 13292   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939   WWalksN cwwlksn 26718

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-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wwlksnexthasheq  26798