Theorem wlksnwwlknvbij 26803

Description: There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length and starting at the same vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 20-Apr-2021.)

Assertion

Ref	Expression
wlksnwwlknvbij	|- ( ( G e. USGraph /\ N e. NN0 /\ X e. (Vtx ` G ) ) -> E. f f : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } )

Distinct variable groups:   f, G, p, w   f, N, p, w   f, X, p, w

Proof of Theorem wlksnwwlknvbij

Dummy variables q s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . 5 |- (Walks ` G ) e. _V
2	1	mptrabex 6488	. . . 4 |- ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) e. _V
3	2	resex 5443	. . 3 |- ( ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) |` { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } ) e. _V
4		eqid 2622	. . . 4 |- ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) = ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) )
5		usgruspgr 26073	. . . . . 6 |- ( G e. USGraph -> G e. USPGraph )
6		fveq2 6191	. . . . . . . . . 10 |- ( q = t -> ( 1st ` q ) = ( 1st ` t ) )
7	6	fveq2d 6195	. . . . . . . . 9 |- ( q = t -> ( # ` ( 1st ` q ) ) = ( # ` ( 1st ` t ) ) )
8	7	eqeq1d 2624	. . . . . . . 8 |- ( q = t -> ( ( # ` ( 1st ` q ) ) = N <-> ( # ` ( 1st ` t ) ) = N ) )
9	8	cbvrabv 3199	. . . . . . 7 |- { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } = { t e. (Walks ` G ) | ( # ` ( 1st ` t ) ) = N }
10		eqid 2622	. . . . . . 7 |- ( N WWalksN G ) = ( N WWalksN G )
11		fveq2 6191	. . . . . . . 8 |- ( p = s -> ( 2nd ` p ) = ( 2nd ` s ) )
12	11	cbvmptv 4750	. . . . . . 7 |- ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) = ( s e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` s ) )
13	9, 10, 12	wlknwwlksnbij 26777	. . . . . 6 |- ( ( G e. USPGraph /\ N e. NN0 ) -> ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) : { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } -1-1-onto-> ( N WWalksN G ) )
14	5, 13	sylan 488	. . . . 5 |- ( ( G e. USGraph /\ N e. NN0 ) -> ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) : { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } -1-1-onto-> ( N WWalksN G ) )
15	14	3adant3 1081	. . . 4 |- ( ( G e. USGraph /\ N e. NN0 /\ X e. (Vtx ` G ) ) -> ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) : { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } -1-1-onto-> ( N WWalksN G ) )
16		fveq1 6190	. . . . . 6 |- ( w = ( 2nd ` p ) -> ( w ` 0 ) = ( ( 2nd ` p ) ` 0 ) )
17	16	eqeq1d 2624	. . . . 5 |- ( w = ( 2nd ` p ) -> ( ( w ` 0 ) = X <-> ( ( 2nd ` p ) ` 0 ) = X ) )
18	17	3ad2ant3 1084	. . . 4 |- ( ( ( G e. USGraph /\ N e. NN0 /\ X e. (Vtx ` G ) ) /\ p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } /\ w = ( 2nd ` p ) ) -> ( ( w ` 0 ) = X <-> ( ( 2nd ` p ) ` 0 ) = X ) )
19	4, 15, 18	f1oresrab 6395	. . 3 |- ( ( G e. USGraph /\ N e. NN0 /\ X e. (Vtx ` G ) ) -> ( ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) |` { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } ) : { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } )
20		f1oeq1 6127	. . . 4 |- ( f = ( ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) |` { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } ) -> ( f : { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } <-> ( ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) |` { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } ) : { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } ) )
21	20	spcegv 3294	. . 3 |- ( ( ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) |` { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } ) e. _V -> ( ( ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } |-> ( 2nd ` p ) ) |` { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } ) : { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } -> E. f f : { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } ) )
22	3, 19, 21	mpsyl 68	. 2 |- ( ( G e. USGraph /\ N e. NN0 /\ X e. (Vtx ` G ) ) -> E. f f : { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } )
23		df-rab 2921	. . . . 5 |- { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) } = { p | ( p e. (Walks ` G ) /\ ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) ) }
24		anass 681	. . . . . . 7 |- ( ( ( p e. (Walks ` G ) /\ ( # ` ( 1st ` p ) ) = N ) /\ ( ( 2nd ` p ) ` 0 ) = X ) <-> ( p e. (Walks ` G ) /\ ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) ) )
25	24	bicomi 214	. . . . . 6 |- ( ( p e. (Walks ` G ) /\ ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) ) <-> ( ( p e. (Walks ` G ) /\ ( # ` ( 1st ` p ) ) = N ) /\ ( ( 2nd ` p ) ` 0 ) = X ) )
26	25	abbii 2739	. . . . 5 |- { p | ( p e. (Walks ` G ) /\ ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) ) } = { p | ( ( p e. (Walks ` G ) /\ ( # ` ( 1st ` p ) ) = N ) /\ ( ( 2nd ` p ) ` 0 ) = X ) }
27		fveq2 6191	. . . . . . . . . . . 12 |- ( q = p -> ( 1st ` q ) = ( 1st ` p ) )
28	27	fveq2d 6195	. . . . . . . . . . 11 |- ( q = p -> ( # ` ( 1st ` q ) ) = ( # ` ( 1st ` p ) ) )
29	28	eqeq1d 2624	. . . . . . . . . 10 |- ( q = p -> ( ( # ` ( 1st ` q ) ) = N <-> ( # ` ( 1st ` p ) ) = N ) )
30	29	elrab 3363	. . . . . . . . 9 |- ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } <-> ( p e. (Walks ` G ) /\ ( # ` ( 1st ` p ) ) = N ) )
31	30	anbi1i 731	. . . . . . . 8 |- ( ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } /\ ( ( 2nd ` p ) ` 0 ) = X ) <-> ( ( p e. (Walks ` G ) /\ ( # ` ( 1st ` p ) ) = N ) /\ ( ( 2nd ` p ) ` 0 ) = X ) )
32	31	bicomi 214	. . . . . . 7 |- ( ( ( p e. (Walks ` G ) /\ ( # ` ( 1st ` p ) ) = N ) /\ ( ( 2nd ` p ) ` 0 ) = X ) <-> ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } /\ ( ( 2nd ` p ) ` 0 ) = X ) )
33	32	abbii 2739	. . . . . 6 |- { p | ( ( p e. (Walks ` G ) /\ ( # ` ( 1st ` p ) ) = N ) /\ ( ( 2nd ` p ) ` 0 ) = X ) } = { p | ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } /\ ( ( 2nd ` p ) ` 0 ) = X ) }
34		df-rab 2921	. . . . . 6 |- { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } = { p | ( p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } /\ ( ( 2nd ` p ) ` 0 ) = X ) }
35	33, 34	eqtr4i 2647	. . . . 5 |- { p | ( ( p e. (Walks ` G ) /\ ( # ` ( 1st ` p ) ) = N ) /\ ( ( 2nd ` p ) ` 0 ) = X ) } = { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X }
36	23, 26, 35	3eqtri 2648	. . . 4 |- { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) } = { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X }
37		f1oeq2 6128	. . . 4 |- ( { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) } = { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } -> ( f : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } <-> f : { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } ) )
38	36, 37	mp1i 13	. . 3 |- ( ( G e. USGraph /\ N e. NN0 /\ X e. (Vtx ` G ) ) -> ( f : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } <-> f : { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } ) )
39	38	exbidv 1850	. 2 |- ( ( G e. USGraph /\ N e. NN0 /\ X e. (Vtx ` G ) ) -> ( E. f f : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } <-> E. f f : { p e. { q e. (Walks ` G ) | ( # ` ( 1st ` q ) ) = N } | ( ( 2nd ` p ) ` 0 ) = X } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } ) )
40	22, 39	mpbird 247	1 |- ( ( G e. USGraph /\ N e. NN0 /\ X e. (Vtx ` G ) ) -> E. f f : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = X ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = X } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   |-> cmpt 4729   |` cres 5116   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   NN0cn0 11292   #chash 13117  Vtxcvtx 25874   USPGraph cuspgr 26043   USGraph cusgr 26044  Walkscwlks 26492   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-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-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-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-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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-usgr 26046  df-wlks 26495  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  rusgrnumwlkg  26872