Theorem wwlksnextprop 26807

Description: Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.)

Hypotheses

Ref	Expression
wwlksnextprop.x	|- X = ( ( N + 1 ) WWalksN G )
wwlksnextprop.e	|- E = (Edg ` G )
wwlksnextprop.y	|- Y = { w e. ( N WWalksN G ) | ( w ` 0 ) = P }

Assertion

Ref	Expression
wwlksnextprop	|- ( N e. NN0 -> { x e. X | ( x ` 0 ) = P } = { x e. X | E. y e. Y ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } )

Distinct variable groups:   w, G   w, N   w, P   y, E   x, N, y   y, P   y, X   y, Y   x, w

Allowed substitution hints:   P( x)   E( x, w)   G( x, y)   X( x, w)   Y( x, w)

Proof of Theorem wwlksnextprop

Step	Hyp	Ref	Expression
1		eqidd 2623	. . . . 5 |- ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) -> ( x substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) )
2		wwlksnextprop.x	. . . . . . . . 9 |- X = ( ( N + 1 ) WWalksN G )
3	2	wwlksnextproplem1 26804	. . . . . . . 8 |- ( ( x e. X /\ N e. NN0 ) -> ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = ( x ` 0 ) )
4	3	ancoms 469	. . . . . . 7 |- ( ( N e. NN0 /\ x e. X ) -> ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = ( x ` 0 ) )
5	4	adantr 481	. . . . . 6 |- ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) -> ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = ( x ` 0 ) )
6		eqeq2 2633	. . . . . . 7 |- ( ( x ` 0 ) = P -> ( ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = ( x ` 0 ) <-> ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = P ) )
7	6	adantl 482	. . . . . 6 |- ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) -> ( ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = ( x ` 0 ) <-> ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = P ) )
8	5, 7	mpbid 222	. . . . 5 |- ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) -> ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = P )
9		wwlksnextprop.e	. . . . . . . 8 |- E = (Edg ` G )
10	2, 9	wwlksnextproplem2 26805	. . . . . . 7 |- ( ( x e. X /\ N e. NN0 ) -> { ( lastS ` ( x substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` x ) } e. E )
11	10	ancoms 469	. . . . . 6 |- ( ( N e. NN0 /\ x e. X ) -> { ( lastS ` ( x substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` x ) } e. E )
12	11	adantr 481	. . . . 5 |- ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) -> { ( lastS ` ( x substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` x ) } e. E )
13		simpr 477	. . . . . . . 8 |- ( ( N e. NN0 /\ x e. X ) -> x e. X )
14	13	adantr 481	. . . . . . 7 |- ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) -> x e. X )
15		simpr 477	. . . . . . 7 |- ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) -> ( x ` 0 ) = P )
16		simpll 790	. . . . . . 7 |- ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) -> N e. NN0 )
17		wwlksnextprop.y	. . . . . . . 8 |- Y = { w e. ( N WWalksN G ) | ( w ` 0 ) = P }
18	2, 9, 17	wwlksnextproplem3 26806	. . . . . . 7 |- ( ( x e. X /\ ( x ` 0 ) = P /\ N e. NN0 ) -> ( x substr <. 0 , ( N + 1 ) >. ) e. Y )
19	14, 15, 16, 18	syl3anc 1326	. . . . . 6 |- ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) -> ( x substr <. 0 , ( N + 1 ) >. ) e. Y )
20		eqeq2 2633	. . . . . . . 8 |- ( y = ( x substr <. 0 , ( N + 1 ) >. ) -> ( ( x substr <. 0 , ( N + 1 ) >. ) = y <-> ( x substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) ) )
21		fveq1 6190	. . . . . . . . 9 |- ( y = ( x substr <. 0 , ( N + 1 ) >. ) -> ( y ` 0 ) = ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) )
22	21	eqeq1d 2624	. . . . . . . 8 |- ( y = ( x substr <. 0 , ( N + 1 ) >. ) -> ( ( y ` 0 ) = P <-> ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = P ) )
23		fveq2 6191	. . . . . . . . . 10 |- ( y = ( x substr <. 0 , ( N + 1 ) >. ) -> ( lastS ` y ) = ( lastS ` ( x substr <. 0 , ( N + 1 ) >. ) ) )
24	23	preq1d 4274	. . . . . . . . 9 |- ( y = ( x substr <. 0 , ( N + 1 ) >. ) -> { ( lastS ` y ) , ( lastS ` x ) } = { ( lastS ` ( x substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` x ) } )
25	24	eleq1d 2686	. . . . . . . 8 |- ( y = ( x substr <. 0 , ( N + 1 ) >. ) -> ( { ( lastS ` y ) , ( lastS ` x ) } e. E <-> { ( lastS ` ( x substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` x ) } e. E ) )
26	20, 22, 25	3anbi123d 1399	. . . . . . 7 |- ( y = ( x substr <. 0 , ( N + 1 ) >. ) -> ( ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) <-> ( ( x substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) /\ ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = P /\ { ( lastS ` ( x substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` x ) } e. E ) ) )
27	26	adantl 482	. . . . . 6 |- ( ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) /\ y = ( x substr <. 0 , ( N + 1 ) >. ) ) -> ( ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) <-> ( ( x substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) /\ ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = P /\ { ( lastS ` ( x substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` x ) } e. E ) ) )
28	19, 27	rspcedv 3313	. . . . 5 |- ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) -> ( ( ( x substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) /\ ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = P /\ { ( lastS ` ( x substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` x ) } e. E ) -> E. y e. Y ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) ) )
29	1, 8, 12, 28	mp3and 1427	. . . 4 |- ( ( ( N e. NN0 /\ x e. X ) /\ ( x ` 0 ) = P ) -> E. y e. Y ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) )
30	29	ex 450	. . 3 |- ( ( N e. NN0 /\ x e. X ) -> ( ( x ` 0 ) = P -> E. y e. Y ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) ) )
31	21	eqcoms 2630	. . . . . . . . 9 |- ( ( x substr <. 0 , ( N + 1 ) >. ) = y -> ( y ` 0 ) = ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) )
32	31	eqeq1d 2624	. . . . . . . 8 |- ( ( x substr <. 0 , ( N + 1 ) >. ) = y -> ( ( y ` 0 ) = P <-> ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = P ) )
33	3	eqcomd 2628	. . . . . . . . . . 11 |- ( ( x e. X /\ N e. NN0 ) -> ( x ` 0 ) = ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) )
34	33	ancoms 469	. . . . . . . . . 10 |- ( ( N e. NN0 /\ x e. X ) -> ( x ` 0 ) = ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) )
35	34	adantr 481	. . . . . . . . 9 |- ( ( ( N e. NN0 /\ x e. X ) /\ y e. Y ) -> ( x ` 0 ) = ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) )
36		eqeq2 2633	. . . . . . . . . 10 |- ( P = ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) -> ( ( x ` 0 ) = P <-> ( x ` 0 ) = ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) ) )
37	36	eqcoms 2630	. . . . . . . . 9 |- ( ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = P -> ( ( x ` 0 ) = P <-> ( x ` 0 ) = ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) ) )
38	35, 37	syl5ibr 236	. . . . . . . 8 |- ( ( ( x substr <. 0 , ( N + 1 ) >. ) ` 0 ) = P -> ( ( ( N e. NN0 /\ x e. X ) /\ y e. Y ) -> ( x ` 0 ) = P ) )
39	32, 38	syl6bi 243	. . . . . . 7 |- ( ( x substr <. 0 , ( N + 1 ) >. ) = y -> ( ( y ` 0 ) = P -> ( ( ( N e. NN0 /\ x e. X ) /\ y e. Y ) -> ( x ` 0 ) = P ) ) )
40	39	imp 445	. . . . . 6 |- ( ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P ) -> ( ( ( N e. NN0 /\ x e. X ) /\ y e. Y ) -> ( x ` 0 ) = P ) )
41	40	3adant3 1081	. . . . 5 |- ( ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) -> ( ( ( N e. NN0 /\ x e. X ) /\ y e. Y ) -> ( x ` 0 ) = P ) )
42	41	com12 32	. . . 4 |- ( ( ( N e. NN0 /\ x e. X ) /\ y e. Y ) -> ( ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) -> ( x ` 0 ) = P ) )
43	42	rexlimdva 3031	. . 3 |- ( ( N e. NN0 /\ x e. X ) -> ( E. y e. Y ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) -> ( x ` 0 ) = P ) )
44	30, 43	impbid 202	. 2 |- ( ( N e. NN0 /\ x e. X ) -> ( ( x ` 0 ) = P <-> E. y e. Y ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) ) )
45	44	rabbidva 3188	1 |- ( N e. NN0 -> { x e. X | ( x ` 0 ) = P } = { x e. X | E. y e. Y ( ( x substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   {cpr 4179   <.cop 4183   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   lastS clsw 13292   substr csubstr 13295  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-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-lsw 13300  df-substr 13303  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by: (None)