Theorem wlkwwlksur 26783

Description: Lemma 3 for wlkwwlkbij2 26785. (Contributed by Alexander van der Vekens, 23-Jul-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
wlkwwlksur	|- ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) -> F : T -onto-> 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   F, p   T, p   W, p

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

Proof of Theorem wlkwwlksur

Dummy variables u f are mutually distinct and 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		fveq1 6190	. . . . . . 7 |- ( w = p -> ( w ` 0 ) = ( p ` 0 ) )
8	7	eqeq1d 2624	. . . . . 6 |- ( w = p -> ( ( w ` 0 ) = P <-> ( p ` 0 ) = P ) )
9	8, 3	elrab2 3366	. . . . 5 |- ( p e. W <-> ( p e. ( N WWalksN G ) /\ ( p ` 0 ) = P ) )
10		wlklnwwlkn 26770	. . . . . . . . . . 11 |- ( G e. USPGraph -> ( E. f ( f (Walks ` G ) p /\ ( # ` f ) = N ) <-> p e. ( N WWalksN G ) ) )
11		df-br 4654	. . . . . . . . . . . . 13 |- ( f (Walks ` G ) p <-> <. f , p >. e. (Walks ` G ) )
12		vex 3203	. . . . . . . . . . . . . . . . 17 |- f e. _V
13		vex 3203	. . . . . . . . . . . . . . . . 17 |- p e. _V
14	12, 13	op1st 7176	. . . . . . . . . . . . . . . 16 |- ( 1st ` <. f , p >. ) = f
15	14	eqcomi 2631	. . . . . . . . . . . . . . 15 |- f = ( 1st ` <. f , p >. )
16	15	fveq2i 6194	. . . . . . . . . . . . . 14 |- ( # ` f ) = ( # ` ( 1st ` <. f , p >. ) )
17	16	eqeq1i 2627	. . . . . . . . . . . . 13 |- ( ( # ` f ) = N <-> ( # ` ( 1st ` <. f , p >. ) ) = N )
18	12, 13	op2nd 7177	. . . . . . . . . . . . . . . . 17 |- ( 2nd ` <. f , p >. ) = p
19	18	eqcomi 2631	. . . . . . . . . . . . . . . 16 |- p = ( 2nd ` <. f , p >. )
20	19	fveq1i 6192	. . . . . . . . . . . . . . 15 |- ( p ` 0 ) = ( ( 2nd ` <. f , p >. ) ` 0 )
21	20	eqeq1i 2627	. . . . . . . . . . . . . 14 |- ( ( p ` 0 ) = P <-> ( ( 2nd ` <. f , p >. ) ` 0 ) = P )
22		opex 4932	. . . . . . . . . . . . . . . 16 |- <. f , p >. e. _V
23	22	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( <. f , p >. e. (Walks ` G ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) -> <. f , p >. e. _V )
24		simpll 790	. . . . . . . . . . . . . . . . . 18 |- ( ( ( <. f , p >. e. (Walks ` G ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) /\ ( ( 2nd ` <. f , p >. ) ` 0 ) = P ) -> <. f , p >. e. (Walks ` G ) )
25		simpr 477	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. f , p >. e. (Walks ` G ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) -> ( # ` ( 1st ` <. f , p >. ) ) = N )
26	25	anim1i 592	. . . . . . . . . . . . . . . . . 18 |- ( ( ( <. f , p >. e. (Walks ` G ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) /\ ( ( 2nd ` <. f , p >. ) ` 0 ) = P ) -> ( ( # ` ( 1st ` <. f , p >. ) ) = N /\ ( ( 2nd ` <. f , p >. ) ` 0 ) = P ) )
27	19	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ( <. f , p >. e. (Walks ` G ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) /\ ( ( 2nd ` <. f , p >. ) ` 0 ) = P ) -> p = ( 2nd ` <. f , p >. ) )
28	24, 26, 27	jca31 557	. . . . . . . . . . . . . . . . 17 |- ( ( ( <. f , p >. e. (Walks ` G ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) /\ ( ( 2nd ` <. f , p >. ) ` 0 ) = P ) -> ( ( <. f , p >. e. (Walks ` G ) /\ ( ( # ` ( 1st ` <. f , p >. ) ) = N /\ ( ( 2nd ` <. f , p >. ) ` 0 ) = P ) ) /\ p = ( 2nd ` <. f , p >. ) ) )
29		eleq1 2689	. . . . . . . . . . . . . . . . . . 19 |- ( u = <. f , p >. -> ( u e. (Walks ` G ) <-> <. f , p >. e. (Walks ` G ) ) )
30		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 |- ( u = <. f , p >. -> ( 1st ` u ) = ( 1st ` <. f , p >. ) )
31	30	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . 21 |- ( u = <. f , p >. -> ( # ` ( 1st ` u ) ) = ( # ` ( 1st ` <. f , p >. ) ) )
32	31	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . 20 |- ( u = <. f , p >. -> ( ( # ` ( 1st ` u ) ) = N <-> ( # ` ( 1st ` <. f , p >. ) ) = N ) )
33		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 |- ( u = <. f , p >. -> ( 2nd ` u ) = ( 2nd ` <. f , p >. ) )
34	33	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . 21 |- ( u = <. f , p >. -> ( ( 2nd ` u ) ` 0 ) = ( ( 2nd ` <. f , p >. ) ` 0 ) )
35	34	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . 20 |- ( u = <. f , p >. -> ( ( ( 2nd ` u ) ` 0 ) = P <-> ( ( 2nd ` <. f , p >. ) ` 0 ) = P ) )
36	32, 35	anbi12d 747	. . . . . . . . . . . . . . . . . . 19 |- ( u = <. f , p >. -> ( ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) <-> ( ( # ` ( 1st ` <. f , p >. ) ) = N /\ ( ( 2nd ` <. f , p >. ) ` 0 ) = P ) ) )
37	29, 36	anbi12d 747	. . . . . . . . . . . . . . . . . 18 |- ( u = <. f , p >. -> ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) <-> ( <. f , p >. e. (Walks ` G ) /\ ( ( # ` ( 1st ` <. f , p >. ) ) = N /\ ( ( 2nd ` <. f , p >. ) ` 0 ) = P ) ) ) )
38	33	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 |- ( u = <. f , p >. -> ( p = ( 2nd ` u ) <-> p = ( 2nd ` <. f , p >. ) ) )
39	37, 38	anbi12d 747	. . . . . . . . . . . . . . . . 17 |- ( u = <. f , p >. -> ( ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) /\ p = ( 2nd ` u ) ) <-> ( ( <. f , p >. e. (Walks ` G ) /\ ( ( # ` ( 1st ` <. f , p >. ) ) = N /\ ( ( 2nd ` <. f , p >. ) ` 0 ) = P ) ) /\ p = ( 2nd ` <. f , p >. ) ) ) )
40	28, 39	syl5ibrcom 237	. . . . . . . . . . . . . . . 16 |- ( ( ( <. f , p >. e. (Walks ` G ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) /\ ( ( 2nd ` <. f , p >. ) ` 0 ) = P ) -> ( u = <. f , p >. -> ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) /\ p = ( 2nd ` u ) ) ) )
41	40	impancom 456	. . . . . . . . . . . . . . 15 |- ( ( ( <. f , p >. e. (Walks ` G ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) /\ u = <. f , p >. ) -> ( ( ( 2nd ` <. f , p >. ) ` 0 ) = P -> ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) /\ p = ( 2nd ` u ) ) ) )
42	23, 41	spcimedv 3292	. . . . . . . . . . . . . 14 |- ( ( <. f , p >. e. (Walks ` G ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) -> ( ( ( 2nd ` <. f , p >. ) ` 0 ) = P -> E. u ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) /\ p = ( 2nd ` u ) ) ) )
43	21, 42	syl5bi 232	. . . . . . . . . . . . 13 |- ( ( <. f , p >. e. (Walks ` G ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) -> ( ( p ` 0 ) = P -> E. u ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) /\ p = ( 2nd ` u ) ) ) )
44	11, 17, 43	syl2anb 496	. . . . . . . . . . . 12 |- ( ( f (Walks ` G ) p /\ ( # ` f ) = N ) -> ( ( p ` 0 ) = P -> E. u ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) /\ p = ( 2nd ` u ) ) ) )
45	44	exlimiv 1858	. . . . . . . . . . 11 |- ( E. f ( f (Walks ` G ) p /\ ( # ` f ) = N ) -> ( ( p ` 0 ) = P -> E. u ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) /\ p = ( 2nd ` u ) ) ) )
46	10, 45	syl6bir 244	. . . . . . . . . 10 |- ( G e. USPGraph -> ( p e. ( N WWalksN G ) -> ( ( p ` 0 ) = P -> E. u ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) /\ p = ( 2nd ` u ) ) ) ) )
47	46	imp32 449	. . . . . . . . 9 |- ( ( G e. USPGraph /\ ( p e. ( N WWalksN G ) /\ ( p ` 0 ) = P ) ) -> E. u ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) /\ p = ( 2nd ` u ) ) )
48		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( p = u -> ( 1st ` p ) = ( 1st ` u ) )
49	48	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( p = u -> ( # ` ( 1st ` p ) ) = ( # ` ( 1st ` u ) ) )
50	49	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( p = u -> ( ( # ` ( 1st ` p ) ) = N <-> ( # ` ( 1st ` u ) ) = N ) )
51		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( p = u -> ( 2nd ` p ) = ( 2nd ` u ) )
52	51	fveq1d 6193	. . . . . . . . . . . . . 14 |- ( p = u -> ( ( 2nd ` p ) ` 0 ) = ( ( 2nd ` u ) ` 0 ) )
53	52	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( p = u -> ( ( ( 2nd ` p ) ` 0 ) = P <-> ( ( 2nd ` u ) ` 0 ) = P ) )
54	50, 53	anbi12d 747	. . . . . . . . . . . 12 |- ( p = u -> ( ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) <-> ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) )
55	54	elrab 3363	. . . . . . . . . . 11 |- ( u e. { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } <-> ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) )
56	55	anbi1i 731	. . . . . . . . . 10 |- ( ( u e. { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } /\ p = ( 2nd ` u ) ) <-> ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) /\ p = ( 2nd ` u ) ) )
57	56	exbii 1774	. . . . . . . . 9 |- ( E. u ( u e. { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } /\ p = ( 2nd ` u ) ) <-> E. u ( ( u e. (Walks ` G ) /\ ( ( # ` ( 1st ` u ) ) = N /\ ( ( 2nd ` u ) ` 0 ) = P ) ) /\ p = ( 2nd ` u ) ) )
58	47, 57	sylibr 224	. . . . . . . 8 |- ( ( G e. USPGraph /\ ( p e. ( N WWalksN G ) /\ ( p ` 0 ) = P ) ) -> E. u ( u e. { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } /\ p = ( 2nd ` u ) ) )
59		df-rex 2918	. . . . . . . 8 |- ( E. u e. { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } p = ( 2nd ` u ) <-> E. u ( u e. { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } /\ p = ( 2nd ` u ) ) )
60	58, 59	sylibr 224	. . . . . . 7 |- ( ( G e. USPGraph /\ ( p e. ( N WWalksN G ) /\ ( p ` 0 ) = P ) ) -> E. u e. { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } p = ( 2nd ` u ) )
61	2	rexeqi 3143	. . . . . . 7 |- ( E. u e. T p = ( 2nd ` u ) <-> E. u e. { p e. (Walks ` G ) | ( ( # ` ( 1st ` p ) ) = N /\ ( ( 2nd ` p ) ` 0 ) = P ) } p = ( 2nd ` u ) )
62	60, 61	sylibr 224	. . . . . 6 |- ( ( G e. USPGraph /\ ( p e. ( N WWalksN G ) /\ ( p ` 0 ) = P ) ) -> E. u e. T p = ( 2nd ` u ) )
63		fveq2 6191	. . . . . . . . 9 |- ( t = u -> ( 2nd ` t ) = ( 2nd ` u ) )
64		fvex 6201	. . . . . . . . 9 |- ( 2nd ` u ) e. _V
65	63, 4, 64	fvmpt 6282	. . . . . . . 8 |- ( u e. T -> ( F ` u ) = ( 2nd ` u ) )
66	65	eqeq2d 2632	. . . . . . 7 |- ( u e. T -> ( p = ( F ` u ) <-> p = ( 2nd ` u ) ) )
67	66	rexbiia 3040	. . . . . 6 |- ( E. u e. T p = ( F ` u ) <-> E. u e. T p = ( 2nd ` u ) )
68	62, 67	sylibr 224	. . . . 5 |- ( ( G e. USPGraph /\ ( p e. ( N WWalksN G ) /\ ( p ` 0 ) = P ) ) -> E. u e. T p = ( F ` u ) )
69	9, 68	sylan2b 492	. . . 4 |- ( ( G e. USPGraph /\ p e. W ) -> E. u e. T p = ( F ` u ) )
70	69	ralrimiva 2966	. . 3 |- ( G e. USPGraph -> A. p e. W E. u e. T p = ( F ` u ) )
71	70	3ad2ant1 1082	. 2 |- ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) -> A. p e. W E. u e. T p = ( F ` u ) )
72		dffo3 6374	. 2 |- ( F : T -onto-> W <-> ( F : T --> W /\ A. p e. W E. u e. T p = ( F ` u ) ) )
73	6, 71, 72	sylanbrc 698	1 |- ( ( G e. USPGraph /\ P e. V /\ N e. NN0 ) -> F : T -onto-> W )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   -onto->wfo 5886   ` 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