Theorem wlkwwlkinj 26782

Description: Lemma 2 for wlkwwlkbij2 26785. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Apr-2021.)

Hypotheses

Ref	Expression
wlkwwlkbij.t	|- T = { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) }
wlkwwlkbij.w	|- W = { w e. ( N WWalksN G ) | ( w ` 0 ) = P }
wlkwwlkbij.f	|- F = ( t e. T |-> ( 2nd ` t ) )

Assertion

Ref	Expression
wlkwwlkinj	|- ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) -> F : T -1-1-> W )

Distinct variable groups:   G, p, t, w   N, p, t, w   P, p, t, w   t, T, w   t, V   t, W   w, F   w, V

Allowed substitution hints:   T( p)   F( t, p)   V( p)   W( w, p)

Proof of Theorem wlkwwlkinj

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgrupgr 26071	. . 3 |- ( G e. USPGraph -> G e. UPGraph )
2		wlkwwlkbij.t	. . . 4 |- T = { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) }
3		wlkwwlkbij.w	. . . 4 |- W = { w e. ( N WWalksN G ) | ( w ` 0 ) = P }
4		wlkwwlkbij.f	. . . 4 |- F = ( t e. T |-> ( 2nd ` t ) )
5	2, 3, 4	wlkwwlkfun 26781	. . 3 |- ( ( G e. UPGraph /\ P e. V /\ N e. NN0 ) -> F : T --> W )
6	1, 5	syl3an1 1359	. 2 |- ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) -> F : T --> W )
7		fveq2 6191	. . . . . . 7 |- ( t = v -> ( 2nd ` t ) = ( 2nd ` v ) )
8		fvex 6201	. . . . . . 7 |- ( 2nd ` v ) e. _V
9	7, 4, 8	fvmpt 6282	. . . . . 6 |- ( v e. T -> ( F ` v ) = ( 2nd ` v ) )
10		fveq2 6191	. . . . . . 7 |- ( t = w -> ( 2nd ` t ) = ( 2nd ` w ) )
11		fvex 6201	. . . . . . 7 |- ( 2nd ` w ) e. _V
12	10, 4, 11	fvmpt 6282	. . . . . 6 |- ( w e. T -> ( F ` w ) = ( 2nd ` w ) )
13	9, 12	eqeqan12d 2638	. . . . 5 |- ( ( v e. T /\ w e. T ) -> ( ( F ` v ) = ( F ` w ) <-> ( 2nd ` v ) = ( 2nd ` w ) ) )
14	13	adantl 482	. . . 4 |- ( ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) /\ ( v e. T /\ w e. T ) ) -> ( ( F ` v ) = ( F ` w ) <-> ( 2nd ` v ) = ( 2nd ` w ) ) )
15		fveq2 6191	. . . . . . . . . 10 |- ( p = v -> ( 1st ` p ) = ( 1st ` v ) )
16	15	fveq2d 6195	. . . . . . . . 9 |- ( p = v -> ( # ` ( 1st ` p ) ) = ( # ` ( 1st ` v ) ) )
17	16	eqeq1d 2624	. . . . . . . 8 |- ( p = v -> ( ( # ` ( 1st ` p ) ) = N <-> ( # ` ( 1st ` v ) ) = N ) )
18		fveq2 6191	. . . . . . . . . 10 |- ( p = v -> ( 2nd ` p ) = ( 2nd ` v ) )
19	18	fveq1d 6193	. . . . . . . . 9 |- ( p = v -> ( ( 2nd ` p ) ` 0 ) = ( ( 2nd ` v ) ` 0 ) )
20	19	eqeq1d 2624	. . . . . . . 8 |- ( p = v -> ( ( ( 2nd ` p ) ` 0 ) = P <-> ( ( 2nd ` v ) ` 0 ) = P ) )
21	17, 20	anbi12d 747	. . . . . . 7 |- ( p = v -> ( ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) <-> ( ( # ` ( 1st ` v ) ) = N /\ ( ( 2nd ` v ) ` 0 ) = P ) ) )
22	21, 2	elrab2 3366	. . . . . 6 |- ( v e. T <-> ( v e. (Walks ` G ) /\ ( ( # ` ( 1st ` v ) ) = N /\ ( ( 2nd ` v ) ` 0 ) = P ) ) )
23		fveq2 6191	. . . . . . . . . 10 |- ( p = w -> ( 1st ` p ) = ( 1st ` w ) )
24	23	fveq2d 6195	. . . . . . . . 9 |- ( p = w -> ( # ` ( 1st ` p ) ) = ( # ` ( 1st ` w ) ) )
25	24	eqeq1d 2624	. . . . . . . 8 |- ( p = w -> ( ( # ` ( 1st ` p ) ) = N <-> ( # ` ( 1st ` w ) ) = N ) )
26		fveq2 6191	. . . . . . . . . 10 |- ( p = w -> ( 2nd ` p ) = ( 2nd ` w ) )
27	26	fveq1d 6193	. . . . . . . . 9 |- ( p = w -> ( ( 2nd ` p ) ` 0 ) = ( ( 2nd ` w ) ` 0 ) )
28	27	eqeq1d 2624	. . . . . . . 8 |- ( p = w -> ( ( ( 2nd ` p ) ` 0 ) = P <-> ( ( 2nd ` w ) ` 0 ) = P ) )
29	25, 28	anbi12d 747	. . . . . . 7 |- ( p = w -> ( ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) <-> ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) ) )
30	29, 2	elrab2 3366	. . . . . 6 |- ( w e. T <-> ( w e. (Walks ` G ) /\ ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) ) )
31	22, 30	anbi12i 733	. . . . 5 |- ( ( v e. T /\ w e. T ) <-> ( ( v e. (Walks ` G ) /\ ( ( # ` ( 1st ` v ) ) = N /\ ( ( 2nd ` v ) ` 0 ) = P ) ) /\ ( w e. (Walks ` G ) /\ ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) ) ) )
32		3simpb 1059	. . . . . . 7 |- ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) -> ( G e. USPGraph /\ N e. NN0 ) )
33	32	adantr 481	. . . . . 6 |- ( ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) /\ ( ( v e. (Walks ` G ) /\ ( ( # ` ( 1st ` v ) ) = N /\ ( ( 2nd ` v ) ` 0 ) = P ) ) /\ ( w e. (Walks ` G ) /\ ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) ) ) ) -> ( G e. USPGraph /\ N e. NN0 ) )
34		simpl 473	. . . . . . . . 9 |- ( ( ( # ` ( 1st ` v ) ) = N /\ ( ( 2nd ` v ) ` 0 ) = P ) -> ( # ` ( 1st ` v ) ) = N )
35	34	anim2i 593	. . . . . . . 8 |- ( ( v e. (Walks ` G ) /\ ( ( # ` ( 1st ` v ) ) = N /\ ( ( 2nd ` v ) ` 0 ) = P ) ) -> ( v e. (Walks ` G ) /\ ( # ` ( 1st ` v ) ) = N ) )
36	35	adantr 481	. . . . . . 7 |- ( ( ( v e. (Walks ` G ) /\ ( ( # ` ( 1st ` v ) ) = N /\ ( ( 2nd ` v ) ` 0 ) = P ) ) /\ ( w e. (Walks ` G ) /\ ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) ) ) -> ( v e. (Walks ` G ) /\ ( # ` ( 1st ` v ) ) = N ) )
37	36	adantl 482	. . . . . 6 |- ( ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) /\ ( ( v e. (Walks ` G ) /\ ( ( # ` ( 1st ` v ) ) = N /\ ( ( 2nd ` v ) ` 0 ) = P ) ) /\ ( w e. (Walks ` G ) /\ ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) ) ) ) -> ( v e. (Walks ` G ) /\ ( # ` ( 1st ` v ) ) = N ) )
38		simpl 473	. . . . . . . . 9 |- ( ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) -> ( # ` ( 1st ` w ) ) = N )
39	38	anim2i 593	. . . . . . . 8 |- ( ( w e. (Walks ` G ) /\ ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) ) -> ( w e. (Walks ` G ) /\ ( # ` ( 1st ` w ) ) = N ) )
40	39	adantl 482	. . . . . . 7 |- ( ( ( v e. (Walks ` G ) /\ ( ( # ` ( 1st ` v ) ) = N /\ ( ( 2nd ` v ) ` 0 ) = P ) ) /\ ( w e. (Walks ` G ) /\ ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) ) ) -> ( w e. (Walks ` G ) /\ ( # ` ( 1st ` w ) ) = N ) )
41	40	adantl 482	. . . . . 6 |- ( ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) /\ ( ( v e. (Walks ` G ) /\ ( ( # ` ( 1st ` v ) ) = N /\ ( ( 2nd ` v ) ` 0 ) = P ) ) /\ ( w e. (Walks ` G ) /\ ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) ) ) ) -> ( w e. (Walks ` G ) /\ ( # ` ( 1st ` w ) ) = N ) )
42		uspgr2wlkeq2 26543	. . . . . 6 |- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( v e. (Walks ` G ) /\ ( # ` ( 1st ` v ) ) = N ) /\ ( w e. (Walks ` G ) /\ ( # ` ( 1st ` w ) ) = N ) ) -> ( ( 2nd ` v ) = ( 2nd ` w ) -> v = w ) )
43	33, 37, 41, 42	syl3anc 1326	. . . . 5 |- ( ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) /\ ( ( v e. (Walks ` G ) /\ ( ( # ` ( 1st ` v ) ) = N /\ ( ( 2nd ` v ) ` 0 ) = P ) ) /\ ( w e. (Walks ` G ) /\ ( ( # ` ( 1st ` w ) ) = N /\ ( ( 2nd ` w ) ` 0 ) = P ) ) ) ) -> ( ( 2nd ` v ) = ( 2nd ` w ) -> v = w ) )
44	31, 43	sylan2b 492	. . . 4 |- ( ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) /\ ( v e. T /\ w e. T ) ) -> ( ( 2nd ` v ) = ( 2nd ` w ) -> v = w ) )
45	14, 44	sylbid 230	. . 3 |- ( ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) /\ ( v e. T /\ w e. T ) ) -> ( ( F ` v ) = ( F ` w ) -> v = w ) )
46	45	ralrimivva 2971	. 2 |- ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) -> A. v e. T A. w e. T ( ( F ` v ) = ( F ` w ) -> v = w ) )
47		dff13 6512	. 2 |- ( F : T -1-1-> W <-> ( F : T --> W /\ A. v e. T A. w e. T ( ( F ` v ) = ( F ` w ) -> v = w ) ) )
48	6, 46, 47	sylanbrc 698	1 |- ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) -> F : T -1-1-> W )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   |-> cmpt 4729   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   NN0cn0 11292   #chash 13117   UPGraph cupgr 25975   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:  wlkwwlkbij  26784