Theorem wwlkextbij 24938

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 5858	. . . 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 4587	. . 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 6117	. . 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 2454	. . . 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 4096	. . . . . 6 |- ( n = p -> { ( lastS ` W ) , n } = { ( lastS ` W ) , p } )
8	7	eleq1d 2523	. . . . 5 |- ( n = p -> ( { ( lastS ` W ) , n } e. ran E <-> { ( lastS ` W ) , p } e. ran E ) )
9	8	cbvrabv 3105	. . . 4 |- { n e. V | { ( lastS ` W ) , n } e. ran E } = { p e. V | { ( lastS ` W ) , p } e. ran E }
10		fveq2 5848	. . . . . . . 8 |- ( t = w -> ( # ` t ) = ( # ` w ) )
11	10	eqeq1d 2456	. . . . . . 7 |- ( t = w -> ( ( # ` t ) = ( N + 2 ) <-> ( # ` w ) = ( N + 2 ) ) )
12		oveq1 6277	. . . . . . . 8 |- ( t = w -> ( t substr <. 0 , ( N + 1 ) >. ) = ( w substr <. 0 , ( N + 1 ) >. ) )
13	12	eqeq1d 2456	. . . . . . 7 |- ( t = w -> ( ( t substr <. 0 , ( N + 1 ) >. ) = W <-> ( w substr <. 0 , ( N + 1 ) >. ) = W ) )
14		fveq2 5848	. . . . . . . . 9 |- ( t = w -> ( lastS ` t ) = ( lastS ` w ) )
15	14	preq2d 4102	. . . . . . . 8 |- ( t = w -> { ( lastS ` W ) , ( lastS ` t ) } = { ( lastS ` W ) , ( lastS ` w ) } )
16	15	eleq1d 2523	. . . . . . 7 |- ( t = w -> ( { ( lastS ` W ) , ( lastS ` t ) } e. ran E <-> { ( lastS ` W ) , ( lastS ` w ) } e. ran E ) )
17	11, 13, 16	3anbi123d 1297	. . . . . 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 3105	. . . . 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 4519	. . . . 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 24937	. . 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 2454	. . . . . . 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 24933	. . . . . 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 2462	. . . . 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 4520	. . . 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 24933	. . . . 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 2462	. . . 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 2455	. . . 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 5795	. . 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 5789	. . 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 3192	. 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 367   /\ w3a 971   = wceq 1398   E.wex 1617   e. wcel 1823   {crab 2808   _Vcvv 3106   {cpr 4018   <.cop 4022   |-> cmpt 4497   ran crn 4989   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482   + caddc 9484   2c2 10581   #chash 12390  Word cword 12521   lastS clsw 12522   substr csubstr 12525   WWalksN cwwlkn 24883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-lsw 12530  df-concat 12531  df-s1 12532  df-substr 12533  df-wwlk 24884  df-wwlkn 24885

This theorem is referenced by:  wwlkexthasheq  24939