Theorem wlknwwlksnsur 26776

Description: Lemma 3 for wlknwwlksnbij2 26778. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)

Hypotheses

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

Assertion

Ref	Expression
wlknwwlksnsur	|- ( ( G e. USPGraph /\ N e. NN0 ) -> F : T -onto-> W )

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

Allowed substitution hint:   F( t)

Proof of Theorem wlknwwlksnsur

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		wlknwwlksnbij.t	. . . 4 |- T = { p e. (Walks ` G ) | ( # ` ( 1st ` p ) ) = N }
3		wlknwwlksnbij.w	. . . 4 |- W = ( N WWalksN G )
4		wlknwwlksnbij.f	. . . 4 |- F = ( t e. T |-> ( 2nd ` t ) )
5	2, 3, 4	wlknwwlksnfun 26774	. . 3 |- ( ( G e. UPGraph /\ N e. NN0 ) -> F : T --> W )
6	1, 5	sylan 488	. 2 |- ( ( G e. USPGraph /\ N e. NN0 ) -> F : T --> W )
7	3	eleq2i 2693	. . . 4 |- ( p e. W <-> p e. ( N WWalksN G ) )
8		wlklnwwlkn 26770	. . . . . . . . . . 11 |- ( G e. USPGraph -> ( E. f ( f (Walks ` G ) p /\ ( # ` f ) = N ) <-> p e. ( N WWalksN G ) ) )
9	8	adantr 481	. . . . . . . . . 10 |- ( ( G e. USPGraph /\ N e. NN0 ) -> ( E. f ( f (Walks ` G ) p /\ ( # ` f ) = N ) <-> p e. ( N WWalksN G ) ) )
10		df-br 4654	. . . . . . . . . . . 12 |- ( f (Walks ` G ) p <-> <. f , p >. e. (Walks ` G ) )
11		vex 3203	. . . . . . . . . . . . . . . 16 |- f e. _V
12		vex 3203	. . . . . . . . . . . . . . . 16 |- p e. _V
13	11, 12	op1st 7176	. . . . . . . . . . . . . . 15 |- ( 1st ` <. f , p >. ) = f
14	13	eqcomi 2631	. . . . . . . . . . . . . 14 |- f = ( 1st ` <. f , p >. )
15	14	fveq2i 6194	. . . . . . . . . . . . 13 |- ( # ` f ) = ( # ` ( 1st ` <. f , p >. ) )
16	15	eqeq1i 2627	. . . . . . . . . . . 12 |- ( ( # ` f ) = N <-> ( # ` ( 1st ` <. f , p >. ) ) = N )
17		elex 3212	. . . . . . . . . . . . . 14 |- ( <. f , p >. e. (Walks ` G ) -> <. f , p >. e. _V )
18		eleq1 2689	. . . . . . . . . . . . . . . . . 18 |- ( u = <. f , p >. -> ( u e. (Walks ` G ) <-> <. f , p >. e. (Walks ` G ) ) )
19	18	biimparc 504	. . . . . . . . . . . . . . . . 17 |- ( ( <. f , p >. e. (Walks ` G ) /\ u = <. f , p >. ) -> u e. (Walks ` G ) )
20	19	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( <. f , p >. e. (Walks ` G ) /\ u = <. f , p >. ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) -> u e. (Walks ` G ) )
21		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( u = <. f , p >. -> ( 1st ` u ) = ( 1st ` <. f , p >. ) )
22	21	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 |- ( u = <. f , p >. -> ( # ` ( 1st ` u ) ) = ( # ` ( 1st ` <. f , p >. ) ) )
23	22	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 |- ( u = <. f , p >. -> ( ( # ` ( 1st ` u ) ) = N <-> ( # ` ( 1st ` <. f , p >. ) ) = N ) )
24	23	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( <. f , p >. e. (Walks ` G ) /\ u = <. f , p >. ) -> ( ( # ` ( 1st ` u ) ) = N <-> ( # ` ( 1st ` <. f , p >. ) ) = N ) )
25	24	biimpar 502	. . . . . . . . . . . . . . . 16 |- ( ( ( <. f , p >. e. (Walks ` G ) /\ u = <. f , p >. ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) -> ( # ` ( 1st ` u ) ) = N )
26		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 |- ( u = <. f , p >. -> ( 2nd ` u ) = ( 2nd ` <. f , p >. ) )
27	11, 12	op2nd 7177	. . . . . . . . . . . . . . . . . . 19 |- ( 2nd ` <. f , p >. ) = p
28	26, 27	syl6req 2673	. . . . . . . . . . . . . . . . . 18 |- ( u = <. f , p >. -> p = ( 2nd ` u ) )
29	28	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( <. f , p >. e. (Walks ` G ) /\ u = <. f , p >. ) -> p = ( 2nd ` u ) )
30	29	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( <. f , p >. e. (Walks ` G ) /\ u = <. f , p >. ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) -> p = ( 2nd ` u ) )
31	20, 25, 30	jca31 557	. . . . . . . . . . . . . . 15 |- ( ( ( <. f , p >. e. (Walks ` G ) /\ u = <. f , p >. ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) -> ( ( u e. (Walks ` G ) /\ ( # ` ( 1st ` u ) ) = N ) /\ p = ( 2nd ` u ) ) )
32	31	ex 450	. . . . . . . . . . . . . 14 |- ( ( <. f , p >. e. (Walks ` G ) /\ u = <. f , p >. ) -> ( ( # ` ( 1st ` <. f , p >. ) ) = N -> ( ( u e. (Walks ` G ) /\ ( # ` ( 1st ` u ) ) = N ) /\ p = ( 2nd ` u ) ) ) )
33	17, 32	spcimedv 3292	. . . . . . . . . . . . 13 |- ( <. f , p >. e. (Walks ` G ) -> ( ( # ` ( 1st ` <. f , p >. ) ) = N -> E. u ( ( u e. (Walks ` G ) /\ ( # ` ( 1st ` u ) ) = N ) /\ p = ( 2nd ` u ) ) ) )
34	33	imp 445	. . . . . . . . . . . 12 |- ( ( <. f , p >. e. (Walks ` G ) /\ ( # ` ( 1st ` <. f , p >. ) ) = N ) -> E. u ( ( u e. (Walks ` G ) /\ ( # ` ( 1st ` u ) ) = N ) /\ p = ( 2nd ` u ) ) )
35	10, 16, 34	syl2anb 496	. . . . . . . . . . 11 |- ( ( f (Walks ` G ) p /\ ( # ` f ) = N ) -> E. u ( ( u e. (Walks ` G ) /\ ( # ` ( 1st ` u ) ) = N ) /\ p = ( 2nd ` u ) ) )
36	35	exlimiv 1858	. . . . . . . . . 10 |- ( E. f ( f (Walks ` G ) p /\ ( # ` f ) = N ) -> E. u ( ( u e. (Walks ` G ) /\ ( # ` ( 1st ` u ) ) = N ) /\ p = ( 2nd ` u ) ) )
37	9, 36	syl6bir 244	. . . . . . . . 9 |- ( ( G e. USPGraph /\ N e. NN0 ) -> ( p e. ( N WWalksN G ) -> E. u ( ( u e. (Walks ` G ) /\ ( # ` ( 1st ` u ) ) = N ) /\ p = ( 2nd ` u ) ) ) )
38	37	imp 445	. . . . . . . 8 |- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ p e. ( N WWalksN G ) ) -> E. u ( ( u e. (Walks ` G ) /\ ( # ` ( 1st ` u ) ) = N ) /\ p = ( 2nd ` u ) ) )
39		fveq2 6191	. . . . . . . . . . . . 13 |- ( p = u -> ( 1st ` p ) = ( 1st ` u ) )
40	39	fveq2d 6195	. . . . . . . . . . . 12 |- ( p = u -> ( # ` ( 1st ` p ) ) = ( # ` ( 1st ` u ) ) )
41	40	eqeq1d 2624	. . . . . . . . . . 11 |- ( p = u -> ( ( # ` ( 1st ` p ) ) = N <-> ( # ` ( 1st ` u ) ) = N ) )
42	41	elrab 3363	. . . . . . . . . 10 |- ( u e. { p e. (Walks ` G ) | ( # ` ( 1st ` p ) ) = N } <-> ( u e. (Walks ` G ) /\ ( # ` ( 1st ` u ) ) = N ) )
43	42	anbi1i 731	. . . . . . . . 9 |- ( ( u e. { p e. (Walks ` G ) | ( # ` ( 1st ` p ) ) = N } /\ p = ( 2nd ` u ) ) <-> ( ( u e. (Walks ` G ) /\ ( # ` ( 1st ` u ) ) = N ) /\ p = ( 2nd ` u ) ) )
44	43	exbii 1774	. . . . . . . 8 |- ( E. u ( u e. { p e. (Walks ` G ) | ( # ` ( 1st ` p ) ) = N } /\ p = ( 2nd ` u ) ) <-> E. u ( ( u e. (Walks ` G ) /\ ( # ` ( 1st ` u ) ) = N ) /\ p = ( 2nd ` u ) ) )
45	38, 44	sylibr 224	. . . . . . 7 |- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ p e. ( N WWalksN G ) ) -> E. u ( u e. { p e. (Walks ` G ) | ( # ` ( 1st ` p ) ) = N } /\ p = ( 2nd ` u ) ) )
46		df-rex 2918	. . . . . . 7 |- ( E. u e. { p e. (Walks ` G ) | ( # ` ( 1st ` p ) ) = N } p = ( 2nd ` u ) <-> E. u ( u e. { p e. (Walks ` G ) | ( # ` ( 1st ` p ) ) = N } /\ p = ( 2nd ` u ) ) )
47	45, 46	sylibr 224	. . . . . 6 |- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ p e. ( N WWalksN G ) ) -> E. u e. { p e. (Walks ` G ) | ( # ` ( 1st ` p ) ) = N } p = ( 2nd ` u ) )
48	2	rexeqi 3143	. . . . . 6 |- ( E. u e. T p = ( 2nd ` u ) <-> E. u e. { p e. (Walks ` G ) | ( # ` ( 1st ` p ) ) = N } p = ( 2nd ` u ) )
49	47, 48	sylibr 224	. . . . 5 |- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ p e. ( N WWalksN G ) ) -> E. u e. T p = ( 2nd ` u ) )
50		fveq2 6191	. . . . . . . 8 |- ( t = u -> ( 2nd ` t ) = ( 2nd ` u ) )
51		fvex 6201	. . . . . . . 8 |- ( 2nd ` u ) e. _V
52	50, 4, 51	fvmpt 6282	. . . . . . 7 |- ( u e. T -> ( F ` u ) = ( 2nd ` u ) )
53	52	eqeq2d 2632	. . . . . 6 |- ( u e. T -> ( p = ( F ` u ) <-> p = ( 2nd ` u ) ) )
54	53	rexbiia 3040	. . . . 5 |- ( E. u e. T p = ( F ` u ) <-> E. u e. T p = ( 2nd ` u ) )
55	49, 54	sylibr 224	. . . 4 |- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ p e. ( N WWalksN G ) ) -> E. u e. T p = ( F ` u ) )
56	7, 55	sylan2b 492	. . 3 |- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ p e. W ) -> E. u e. T p = ( F ` u ) )
57	56	ralrimiva 2966	. 2 |- ( ( G e. USPGraph /\ N e. NN0 ) -> A. p e. W E. u e. T p = ( F ` u ) )
58		dffo3 6374	. 2 |- ( F : T -onto-> W <-> ( F : T --> W /\ A. p e. W E. u e. T p = ( F ` u ) ) )
59	6, 57, 58	sylanbrc 698	1 |- ( ( G e. USPGraph /\ N e. NN0 ) -> F : T -onto-> W )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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   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:  wlknwwlksnbij  26777