Theorem upgr2wlk 26564

Description: Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 28-Oct-2021.)

Hypotheses

Ref	Expression
upgr2wlk.v	|- V = (Vtx ` G )
upgr2wlk.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
upgr2wlk	|- ( G e. UPGraph -> ( ( F (Walks ` G ) P /\ ( # ` F ) = 2 ) <-> ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )

Proof of Theorem upgr2wlk

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgr2wlk.v	. . . 4 |- V = (Vtx ` G )
2		upgr2wlk.i	. . . 4 |- I = (iEdg ` G )
3	1, 2	upgriswlk 26537	. . 3 |- ( G e. UPGraph -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
4	3	anbi1d 741	. 2 |- ( G e. UPGraph -> ( ( F (Walks ` G ) P /\ ( # ` F ) = 2 ) <-> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) /\ ( # ` F ) = 2 ) ) )
5		iswrdb 13311	. . . . . . . . 9 |- ( F e. Word dom I <-> F : ( 0..^ ( # ` F ) ) --> dom I )
6		oveq2 6658	. . . . . . . . . 10 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = ( 0..^ 2 ) )
7	6	feq2d 6031	. . . . . . . . 9 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom I <-> F : ( 0..^ 2 ) --> dom I ) )
8	5, 7	syl5bb 272	. . . . . . . 8 |- ( ( # ` F ) = 2 -> ( F e. Word dom I <-> F : ( 0..^ 2 ) --> dom I ) )
9		oveq2 6658	. . . . . . . . 9 |- ( ( # ` F ) = 2 -> ( 0 ... ( # ` F ) ) = ( 0 ... 2 ) )
10	9	feq2d 6031	. . . . . . . 8 |- ( ( # ` F ) = 2 -> ( P : ( 0 ... ( # ` F ) ) --> V <-> P : ( 0 ... 2 ) --> V ) )
11		fzo0to2pr 12553	. . . . . . . . . . 11 |- ( 0..^ 2 ) = { 0 , 1 }
12	6, 11	syl6eq 2672	. . . . . . . . . 10 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = { 0 , 1 } )
13	12	raleqdv 3144	. . . . . . . . 9 |- ( ( # ` F ) = 2 -> ( A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> A. k e. { 0 , 1 } ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
14		2wlklem 26563	. . . . . . . . 9 |- ( A. k e. { 0 , 1 } ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
15	13, 14	syl6bb 276	. . . . . . . 8 |- ( ( # ` F ) = 2 -> ( A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) )
16	8, 10, 15	3anbi123d 1399	. . . . . . 7 |- ( ( # ` F ) = 2 -> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) <-> ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
17	16	adantl 482	. . . . . 6 |- ( ( G e. UPGraph /\ ( # ` F ) = 2 ) -> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) <-> ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
18		3anass 1042	. . . . . 6 |- ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) <-> ( F : ( 0..^ 2 ) --> dom I /\ ( P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
19	17, 18	syl6bb 276	. . . . 5 |- ( ( G e. UPGraph /\ ( # ` F ) = 2 ) -> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) <-> ( F : ( 0..^ 2 ) --> dom I /\ ( P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) ) )
20	19	ex 450	. . . 4 |- ( G e. UPGraph -> ( ( # ` F ) = 2 -> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) <-> ( F : ( 0..^ 2 ) --> dom I /\ ( P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) ) ) )
21	20	pm5.32rd 672	. . 3 |- ( G e. UPGraph -> ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) /\ ( # ` F ) = 2 ) <-> ( ( F : ( 0..^ 2 ) --> dom I /\ ( P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) /\ ( # ` F ) = 2 ) ) )
22		3anass 1042	. . . 4 |- ( ( ( F : ( 0..^ 2 ) --> dom I /\ ( # ` F ) = 2 ) /\ P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) <-> ( ( F : ( 0..^ 2 ) --> dom I /\ ( # ` F ) = 2 ) /\ ( P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
23		an32 839	. . . 4 |- ( ( ( F : ( 0..^ 2 ) --> dom I /\ ( # ` F ) = 2 ) /\ ( P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) <-> ( ( F : ( 0..^ 2 ) --> dom I /\ ( P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) /\ ( # ` F ) = 2 ) )
24	22, 23	bitri 264	. . 3 |- ( ( ( F : ( 0..^ 2 ) --> dom I /\ ( # ` F ) = 2 ) /\ P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) <-> ( ( F : ( 0..^ 2 ) --> dom I /\ ( P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) /\ ( # ` F ) = 2 ) )
25	21, 24	syl6bbr 278	. 2 |- ( G e. UPGraph -> ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) /\ ( # ` F ) = 2 ) <-> ( ( F : ( 0..^ 2 ) --> dom I /\ ( # ` F ) = 2 ) /\ P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
26		2nn0 11309	. . . . . . 7 |- 2 e. NN0
27		fnfzo0hash 13234	. . . . . . 7 |- ( ( 2 e. NN0 /\ F : ( 0..^ 2 ) --> dom I ) -> ( # ` F ) = 2 )
28	26, 27	mpan 706	. . . . . 6 |- ( F : ( 0..^ 2 ) --> dom I -> ( # ` F ) = 2 )
29	28	pm4.71i 664	. . . . 5 |- ( F : ( 0..^ 2 ) --> dom I <-> ( F : ( 0..^ 2 ) --> dom I /\ ( # ` F ) = 2 ) )
30	29	bicomi 214	. . . 4 |- ( ( F : ( 0..^ 2 ) --> dom I /\ ( # ` F ) = 2 ) <-> F : ( 0..^ 2 ) --> dom I )
31	30	a1i 11	. . 3 |- ( G e. UPGraph -> ( ( F : ( 0..^ 2 ) --> dom I /\ ( # ` F ) = 2 ) <-> F : ( 0..^ 2 ) --> dom I ) )
32	31	3anbi1d 1403	. 2 |- ( G e. UPGraph -> ( ( ( F : ( 0..^ 2 ) --> dom I /\ ( # ` F ) = 2 ) /\ P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) <-> ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
33	4, 25, 32	3bitrd 294	1 |- ( G e. UPGraph -> ( ( F (Walks ` G ) P /\ ( # ` F ) = 2 ) <-> ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {cpr 4179   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975  Walkscwlks 26492

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-wlks 26495

This theorem is referenced by:  umgrwwlks2on  26850