Theorem wlkwwlkbij2 26785

Description: There is a bijection between the set of walks of a fixed length, starting at a fixed vertex, and the set of walks represented as words of the same length, starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 16-Apr-2021.)

Assertion

Ref	Expression
wlkwwlkbij2	|- ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) -> E. f f : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = P } )

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

Allowed substitution hints:   V( w, f, p)

Proof of Theorem wlkwwlkbij2

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

Step	Hyp	Ref	Expression
1		fvex 6201	. . 3 |- (Walks ` G ) e. _V
2	1	mptrabex 6488	. 2 |- ( x e. { t e. (Walks ` G ) | ( ( # ` ( 1st ` t ) ) = N /\ ( ( 2nd ` t ) ` 0 ) = P ) } |-> ( 2nd ` x ) ) e. _V
3		fveq2 6191	. . . . . . 7 |- ( p = u -> ( 1st ` p ) = ( 1st ` u ) )
4	3	fveq2d 6195	. . . . . 6 |- ( p = u -> ( # ` ( 1st ` p ) ) = ( # ` ( 1st ` u ) ) )
5	4	eqeq1d 2624	. . . . 5 |- ( p = u -> ( ( # ` ( 1st ` p ) ) = N <-> ( # ` ( 1st ` u ) ) = N ) )
6		fveq2 6191	. . . . . . 7 |- ( p = u -> ( 2nd ` p ) = ( 2nd ` u ) )
7	6	fveq1d 6193	. . . . . 6 |- ( p = u -> ( ( 2nd ` p ) ` 0 ) = ( ( 2nd ` u ) ` 0 ) )
8	7	eqeq1d 2624	. . . . 5 |- ( p = u -> ( ( ( 2nd ` p ) ` 0 ) = P <-> ( ( 2nd ` u ) ` 0 ) = P ) )
9	5, 8	anbi12d 747	. . . 4 |- ( p = u -> ( ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) <-> ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) )
10	9	cbvrabv 3199	. . 3 |- { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } = { u e. (Walks ` G ) | ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) }
11		fveq1 6190	. . . . 5 |- ( w = s -> ( w ` 0 ) = ( s ` 0 ) )
12	11	eqeq1d 2624	. . . 4 |- ( w = s -> ( ( w ` 0 ) = P <-> ( s ` 0 ) = P ) )
13	12	cbvrabv 3199	. . 3 |- { w e. ( N WWalksN G ) | ( w ` 0 ) = P } = { s e. ( N WWalksN G ) | ( s ` 0 ) = P }
14		fveq2 6191	. . . . . . . 8 |- ( t = p -> ( 1st ` t ) = ( 1st ` p ) )
15	14	fveq2d 6195	. . . . . . 7 |- ( t = p -> ( # ` ( 1st ` t ) ) = ( # ` ( 1st ` p ) ) )
16	15	eqeq1d 2624	. . . . . 6 |- ( t = p -> ( ( # ` ( 1st ` t ) ) = N <-> ( # ` ( 1st ` p ) ) = N ) )
17		fveq2 6191	. . . . . . . 8 |- ( t = p -> ( 2nd ` t ) = ( 2nd ` p ) )
18	17	fveq1d 6193	. . . . . . 7 |- ( t = p -> ( ( 2nd ` t ) ` 0 ) = ( ( 2nd ` p ) ` 0 ) )
19	18	eqeq1d 2624	. . . . . 6 |- ( t = p -> ( ( ( 2nd ` t ) ` 0 ) = P <-> ( ( 2nd ` p ) ` 0 ) = P ) )
20	16, 19	anbi12d 747	. . . . 5 |- ( t = p -> ( ( ( # ` ( 1st ` t ) ) = N /\ ( ( 2nd ` t ) ` 0 ) = P ) <-> ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) ) )
21	20	cbvrabv 3199	. . . 4 |- { t e. (Walks ` G ) | ( ( # ` ( 1st ` t ) ) = N /\ ( ( 2nd ` t ) ` 0 ) = P ) } = { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) }
22	21	mpteq1i 4739	. . 3 |- ( x e. { t e. (Walks ` G ) | ( ( # ` ( 1st ` t ) ) = N /\ ( ( 2nd ` t ) ` 0 ) = P ) } |-> ( 2nd ` x ) ) = ( x e. { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } |-> ( 2nd ` x ) )
23	10, 13, 22	wlkwwlkbij 26784	. 2 |- ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) -> ( x e. { t e. (Walks ` G ) | ( ( # ` ( 1st ` t ) ) = N /\ ( ( 2nd ` t ) ` 0 ) = P ) } |-> ( 2nd ` x ) ) : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = P } )
24		f1oeq1 6127	. . 3 |- ( f = ( x e. { t e. (Walks ` G ) | ( ( # ` ( 1st ` t ) ) = N /\ ( ( 2nd ` t ) ` 0 ) = P ) } |-> ( 2nd ` x ) ) -> ( f : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = P } <-> ( x e. { t e. (Walks ` G ) | ( ( # ` ( 1st ` t ) ) = N /\ ( ( 2nd ` t ) ` 0 ) = P ) } |-> ( 2nd ` x ) ) : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = P } ) )
25	24	spcegv 3294	. 2 |- ( ( x e. { t e. (Walks ` G ) | ( ( # ` ( 1st ` t ) ) = N /\ ( ( 2nd ` t ) ` 0 ) = P ) } |-> ( 2nd ` x ) ) e. _V -> ( ( x e. { t e. (Walks ` G ) | ( ( # ` ( 1st ` t ) ) = N /\ ( ( 2nd ` t ) ` 0 ) = P ) } |-> ( 2nd ` x ) ) : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = P } -> E. f f : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = P } ) )
26	2, 23, 25	mpsyl 68	1 |- ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) -> E. f f : { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } -1-1-onto-> { w e. ( N WWalksN G ) | ( w ` 0 ) = P } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {crab 2916   _Vcvv 3200   |-> cmpt 4729   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   NN0cn0 11292   #chash 13117   USPGraph cuspgr 26043  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-wlks 26495  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by: (None)