Theorem wlkpwwlkf1ouspgr 26765

Description: The mapping of (ordinary) walks to their sequences of vertices is a bijection in a simple pseudograph. (Contributed by AV, 6-May-2021.)

Hypothesis

Ref	Expression
wlkpwwlkf1ouspgr.f	|- F = ( w e. (Walks ` G ) |-> ( 2nd ` w ) )

Assertion

Ref	Expression
wlkpwwlkf1ouspgr	|- ( G e. USPGraph -> F : (Walks ` G ) -1-1-onto-> (WWalks ` G ) )

Distinct variable group:   w, G

Allowed substitution hint:   F( w)

Proof of Theorem wlkpwwlkf1ouspgr

Dummy variables x y f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . . 6 |- ( 1st ` w ) e. _V
2		breq1 4656	. . . . . 6 |- ( f = ( 1st ` w ) -> ( f (Walks ` G ) ( 2nd ` w ) <-> ( 1st ` w ) (Walks ` G ) ( 2nd ` w ) ) )
3	1, 2	spcev 3300	. . . . 5 |- ( ( 1st ` w ) (Walks ` G ) ( 2nd ` w ) -> E. f f (Walks ` G ) ( 2nd ` w ) )
4		wlkiswwlks 26762	. . . . 5 |- ( G e. USPGraph -> ( E. f f (Walks ` G ) ( 2nd ` w ) <-> ( 2nd ` w ) e. (WWalks ` G ) ) )
5	3, 4	syl5ib 234	. . . 4 |- ( G e. USPGraph -> ( ( 1st ` w ) (Walks ` G ) ( 2nd ` w ) -> ( 2nd ` w ) e. (WWalks ` G ) ) )
6		wlkcpr 26524	. . . . 5 |- ( w e. (Walks ` G ) <-> ( 1st ` w ) (Walks ` G ) ( 2nd ` w ) )
7	6	biimpi 206	. . . 4 |- ( w e. (Walks ` G ) -> ( 1st ` w ) (Walks ` G ) ( 2nd ` w ) )
8	5, 7	impel 485	. . 3 |- ( ( G e. USPGraph /\ w e. (Walks ` G ) ) -> ( 2nd ` w ) e. (WWalks ` G ) )
9		wlkpwwlkf1ouspgr.f	. . 3 |- F = ( w e. (Walks ` G ) |-> ( 2nd ` w ) )
10	8, 9	fmptd 6385	. 2 |- ( G e. USPGraph -> F : (Walks ` G ) --> (WWalks ` G ) )
11		simpr 477	. . . 4 |- ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) -> F : (Walks ` G ) --> (WWalks ` G ) )
12	9	a1i 11	. . . . . . . . 9 |- ( x e. (Walks ` G ) -> F = ( w e. (Walks ` G ) |-> ( 2nd ` w ) ) )
13		fveq2 6191	. . . . . . . . . 10 |- ( w = x -> ( 2nd ` w ) = ( 2nd ` x ) )
14	13	adantl 482	. . . . . . . . 9 |- ( ( x e. (Walks ` G ) /\ w = x ) -> ( 2nd ` w ) = ( 2nd ` x ) )
15		id 22	. . . . . . . . 9 |- ( x e. (Walks ` G ) -> x e. (Walks ` G ) )
16		fvexd 6203	. . . . . . . . 9 |- ( x e. (Walks ` G ) -> ( 2nd ` x ) e. _V )
17	12, 14, 15, 16	fvmptd 6288	. . . . . . . 8 |- ( x e. (Walks ` G ) -> ( F ` x ) = ( 2nd ` x ) )
18	9	a1i 11	. . . . . . . . 9 |- ( y e. (Walks ` G ) -> F = ( w e. (Walks ` G ) |-> ( 2nd ` w ) ) )
19		fveq2 6191	. . . . . . . . . 10 |- ( w = y -> ( 2nd ` w ) = ( 2nd ` y ) )
20	19	adantl 482	. . . . . . . . 9 |- ( ( y e. (Walks ` G ) /\ w = y ) -> ( 2nd ` w ) = ( 2nd ` y ) )
21		id 22	. . . . . . . . 9 |- ( y e. (Walks ` G ) -> y e. (Walks ` G ) )
22		fvexd 6203	. . . . . . . . 9 |- ( y e. (Walks ` G ) -> ( 2nd ` y ) e. _V )
23	18, 20, 21, 22	fvmptd 6288	. . . . . . . 8 |- ( y e. (Walks ` G ) -> ( F ` y ) = ( 2nd ` y ) )
24	17, 23	eqeqan12d 2638	. . . . . . 7 |- ( ( x e. (Walks ` G ) /\ y e. (Walks ` G ) ) -> ( ( F ` x ) = ( F ` y ) <-> ( 2nd ` x ) = ( 2nd ` y ) ) )
25	24	adantl 482	. . . . . 6 |- ( ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) /\ ( x e. (Walks ` G ) /\ y e. (Walks ` G ) ) ) -> ( ( F ` x ) = ( F ` y ) <-> ( 2nd ` x ) = ( 2nd ` y ) ) )
26		uspgr2wlkeqi 26544	. . . . . . . 8 |- ( ( G e. USPGraph /\ ( x e. (Walks ` G ) /\ y e. (Walks ` G ) ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) -> x = y )
27	26	ad4ant134 1296	. . . . . . 7 |- ( ( ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) /\ ( x e. (Walks ` G ) /\ y e. (Walks ` G ) ) ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) -> x = y )
28	27	ex 450	. . . . . 6 |- ( ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) /\ ( x e. (Walks ` G ) /\ y e. (Walks ` G ) ) ) -> ( ( 2nd ` x ) = ( 2nd ` y ) -> x = y ) )
29	25, 28	sylbid 230	. . . . 5 |- ( ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) /\ ( x e. (Walks ` G ) /\ y e. (Walks ` G ) ) ) -> ( ( F ` x ) = ( F ` y ) -> x = y ) )
30	29	ralrimivva 2971	. . . 4 |- ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) -> A. x e. (Walks ` G ) A. y e. (Walks ` G ) ( ( F ` x ) = ( F ` y ) -> x = y ) )
31		dff13 6512	. . . 4 |- ( F : (Walks ` G ) -1-1-> (WWalks ` G ) <-> ( F : (Walks ` G ) --> (WWalks ` G ) /\ A. x e. (Walks ` G ) A. y e. (Walks ` G ) ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
32	11, 30, 31	sylanbrc 698	. . 3 |- ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) -> F : (Walks ` G ) -1-1-> (WWalks ` G ) )
33		wlkiswwlks 26762	. . . . . . . . . 10 |- ( G e. USPGraph -> ( E. f f (Walks ` G ) y <-> y e. (WWalks ` G ) ) )
34	33	adantr 481	. . . . . . . . 9 |- ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) -> ( E. f f (Walks ` G ) y <-> y e. (WWalks ` G ) ) )
35		df-br 4654	. . . . . . . . . . 11 |- ( f (Walks ` G ) y <-> <. f , y >. e. (Walks ` G ) )
36		vex 3203	. . . . . . . . . . . . . 14 |- f e. _V
37		vex 3203	. . . . . . . . . . . . . 14 |- y e. _V
38	36, 37	op2nd 7177	. . . . . . . . . . . . 13 |- ( 2nd ` <. f , y >. ) = y
39	38	eqcomi 2631	. . . . . . . . . . . 12 |- y = ( 2nd ` <. f , y >. )
40		opex 4932	. . . . . . . . . . . . 13 |- <. f , y >. e. _V
41		eleq1 2689	. . . . . . . . . . . . . 14 |- ( x = <. f , y >. -> ( x e. (Walks ` G ) <-> <. f , y >. e. (Walks ` G ) ) )
42		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( x = <. f , y >. -> ( 2nd ` x ) = ( 2nd ` <. f , y >. ) )
43	42	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( x = <. f , y >. -> ( y = ( 2nd ` x ) <-> y = ( 2nd ` <. f , y >. ) ) )
44	41, 43	anbi12d 747	. . . . . . . . . . . . 13 |- ( x = <. f , y >. -> ( ( x e. (Walks ` G ) /\ y = ( 2nd ` x ) ) <-> ( <. f , y >. e. (Walks ` G ) /\ y = ( 2nd ` <. f , y >. ) ) ) )
45	40, 44	spcev 3300	. . . . . . . . . . . 12 |- ( ( <. f , y >. e. (Walks ` G ) /\ y = ( 2nd ` <. f , y >. ) ) -> E. x ( x e. (Walks ` G ) /\ y = ( 2nd ` x ) ) )
46	39, 45	mpan2 707	. . . . . . . . . . 11 |- ( <. f , y >. e. (Walks ` G ) -> E. x ( x e. (Walks ` G ) /\ y = ( 2nd ` x ) ) )
47	35, 46	sylbi 207	. . . . . . . . . 10 |- ( f (Walks ` G ) y -> E. x ( x e. (Walks ` G ) /\ y = ( 2nd ` x ) ) )
48	47	exlimiv 1858	. . . . . . . . 9 |- ( E. f f (Walks ` G ) y -> E. x ( x e. (Walks ` G ) /\ y = ( 2nd ` x ) ) )
49	34, 48	syl6bir 244	. . . . . . . 8 |- ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) -> ( y e. (WWalks ` G ) -> E. x ( x e. (Walks ` G ) /\ y = ( 2nd ` x ) ) ) )
50	49	imp 445	. . . . . . 7 |- ( ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) /\ y e. (WWalks ` G ) ) -> E. x ( x e. (Walks ` G ) /\ y = ( 2nd ` x ) ) )
51		df-rex 2918	. . . . . . 7 |- ( E. x e. (Walks ` G ) y = ( 2nd ` x ) <-> E. x ( x e. (Walks ` G ) /\ y = ( 2nd ` x ) ) )
52	50, 51	sylibr 224	. . . . . 6 |- ( ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) /\ y e. (WWalks ` G ) ) -> E. x e. (Walks ` G ) y = ( 2nd ` x ) )
53	17	eqeq2d 2632	. . . . . . 7 |- ( x e. (Walks ` G ) -> ( y = ( F ` x ) <-> y = ( 2nd ` x ) ) )
54	53	rexbiia 3040	. . . . . 6 |- ( E. x e. (Walks ` G ) y = ( F ` x ) <-> E. x e. (Walks ` G ) y = ( 2nd ` x ) )
55	52, 54	sylibr 224	. . . . 5 |- ( ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) /\ y e. (WWalks ` G ) ) -> E. x e. (Walks ` G ) y = ( F ` x ) )
56	55	ralrimiva 2966	. . . 4 |- ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) -> A. y e. (WWalks ` G ) E. x e. (Walks ` G ) y = ( F ` x ) )
57		dffo3 6374	. . . 4 |- ( F : (Walks ` G ) -onto-> (WWalks ` G ) <-> ( F : (Walks ` G ) --> (WWalks ` G ) /\ A. y e. (WWalks ` G ) E. x e. (Walks ` G ) y = ( F ` x ) ) )
58	11, 56, 57	sylanbrc 698	. . 3 |- ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) -> F : (Walks ` G ) -onto-> (WWalks ` G ) )
59		df-f1o 5895	. . 3 |- ( F : (Walks ` G ) -1-1-onto-> (WWalks ` G ) <-> ( F : (Walks ` G ) -1-1-> (WWalks ` G ) /\ F : (Walks ` G ) -onto-> (WWalks ` G ) ) )
60	32, 58, 59	sylanbrc 698	. 2 |- ( ( G e. USPGraph /\ F : (Walks ` G ) --> (WWalks ` G ) ) -> F : (Walks ` G ) -1-1-onto-> (WWalks ` G ) )
61	10, 60	mpdan 702	1 |- ( G e. USPGraph -> F : (Walks ` G ) -1-1-onto-> (WWalks ` G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   USPGraph cuspgr 26043  Walkscwlks 26492  WWalkscwwlks 26717

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

This theorem is referenced by:  wlkisowwlkupgr  26766