Theorem wwlkextbij 30505

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.)

Assertion

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

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

Allowed substitution hint:   N( n)

Proof of Theorem wwlkextbij

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

Step	Hyp	Ref	Expression
1		fvex 5801	. . . 4 |- ( ( V WWalksN E ) ` ( N + 1 ) ) e. _V
2	1	a1i 11	. . 3 |- ( W e. ( ( V WWalksN E ) ` N ) -> ( ( V WWalksN E ) ` ( N + 1 ) ) e. _V )
3		rabexg 4542	. . 3 |- ( ( ( V WWalksN E ) ` ( N + 1 ) ) e. _V -> { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } e. _V )
4		mptexg 6048	. . 3 |- ( { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } e. _V -> ( x e. { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) e. _V )
5	2, 3, 4	3syl 20	. 2 |- ( W e. ( ( V WWalksN E ) ` N ) -> ( x e. { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) e. _V )
6		eqid 2451	. . . 4 |- { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) }
7		preq2 4055	. . . . . 6 |- ( n = p -> { ( lastS ` W ) , n } = { ( lastS ` W ) , p } )
8	7	eleq1d 2520	. . . . 5 |- ( n = p -> ( { ( lastS ` W ) , n } e. ran E <-> { ( lastS ` W ) , p } e. ran E ) )
9	8	cbvrabv 3069	. . . 4 |- { n e. V | { ( lastS ` W ) , n } e. ran E } = { p e. V | { ( lastS ` W ) , p } e. ran E }
10		fveq2 5791	. . . . . . . 8 |- ( t = w -> ( # ` t ) = ( # ` w ) )
11	10	eqeq1d 2453	. . . . . . 7 |- ( t = w -> ( ( # ` t ) = ( N + 2 ) <-> ( # ` w ) = ( N + 2 ) ) )
12		oveq1 6199	. . . . . . . 8 |- ( t = w -> ( t substr <. 0 , ( N + 1 ) >. ) = ( w substr <. 0 , ( N + 1 ) >. ) )
13	12	eqeq1d 2453	. . . . . . 7 |- ( t = w -> ( ( t substr <. 0 , ( N + 1 ) >. ) = W <-> ( w substr <. 0 , ( N + 1 ) >. ) = W ) )
14		fveq2 5791	. . . . . . . . 9 |- ( t = w -> ( lastS ` t ) = ( lastS ` w ) )
15	14	preq2d 4061	. . . . . . . 8 |- ( t = w -> { ( lastS ` W ) , ( lastS ` t ) } = { ( lastS ` W ) , ( lastS ` w ) } )
16	15	eleq1d 2520	. . . . . . 7 |- ( t = w -> ( { ( lastS ` W ) , ( lastS ` t ) } e. ran E <-> { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) )
17	11, 13, 16	3anbi123d 1290	. . . . . 6 |- ( t = w -> ( ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) <-> ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) ) )
18	17	cbvrabv 3069	. . . . 5 |- { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) }
19		mpteq1 4472	. . . . 5 |- ( { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } -> ( x e. { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) = ( x e. { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } |-> ( lastS ` x ) ) )
20	18, 19	ax-mp 5	. . . 4 |- ( x e. { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) = ( x e. { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } |-> ( lastS ` x ) )
21	6, 9, 20	wwlkextbij0 30504	. . 3 |- ( W e. ( ( V WWalksN E ) ` N ) -> ( x e. { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) : { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. ran E } )
22		eqid 2451	. . . . . . 7 |- { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } = { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) }
23	22	wwlkextwrd 30500	. . . . . 6 |- ( W e. ( ( V WWalksN E ) ` N ) -> { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } = { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } )
24	23	eqcomd 2459	. . . . 5 |- ( W e. ( ( V WWalksN E ) ` N ) -> { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } = { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } )
25	24	mpteq1d 4473	. . . 4 |- ( W e. ( ( V WWalksN E ) ` N ) -> ( x e. { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) = ( x e. { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) )
26	6	wwlkextwrd 30500	. . . . 5 |- ( W e. ( ( V WWalksN E ) ` N ) -> { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } = { w e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } )
27	26	eqcomd 2459	. . . 4 |- ( W e. ( ( V WWalksN E ) ` N ) -> { w e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } )
28		eqidd 2452	. . . 4 |- ( W e. ( ( V WWalksN E ) ` N ) -> { n e. V | { ( lastS ` W ) , n } e. ran E } = { n e. V | { ( lastS ` W ) , n } e. ran E } )
29	25, 27, 28	f1oeq123d 5738	. . 3 |- ( W e. ( ( V WWalksN E ) ` N ) -> ( ( x e. { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) : { w e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. ran E } <-> ( x e. { t e. Word V | ( ( # ` t ) = ( N + 2 ) /\ ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) : { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. ran E } ) )
30	21, 29	mpbird 232	. 2 |- ( W e. ( ( V WWalksN E ) ` N ) -> ( x e. { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) : { w e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. ran E } )
31		f1oeq1 5732	. . 3 |- ( f = ( x e. { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) -> ( f : { w e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. ran E } <-> ( x e. { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) : { w e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. ran E } ) )
32	31	spcegv 3156	. 2 |- ( ( x e. { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) e. _V -> ( ( x e. { t e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( t substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` t ) } e. ran E ) } |-> ( lastS ` x ) ) : { w e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. ran E } -> E. f f : { w e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. ran E } ) )
33	5, 30, 32	sylc 60	1 |- ( W e. ( ( V WWalksN E ) ` N ) -> E. f f : { w e. ( ( V WWalksN E ) ` ( N + 1 ) ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) } -1-1-onto-> { n e. V | { ( lastS ` W ) , n } e. ran E } )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   E.wex 1587   e. wcel 1758   {crab 2799   _Vcvv 3070   {cpr 3979   <.cop 3983   |-> cmpt 4450   ran crn 4941   -1-1-onto->wf1o 5517   ` cfv 5518  (class class class)co 6192   0cc0 9385   1c1 9386   + caddc 9388   2c2 10474   #chash 12206  Word cword 12325   lastS clsw 12326   substr csubstr 12329   WWalksN cwwlkn 30452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-hash 12207  df-word 12333  df-lsw 12334  df-concat 12335  df-s1 12336  df-substr 12337  df-wwlk 30453  df-wwlkn 30454

This theorem is referenced by:  wwlkexthasheq  30506