Theorem isupwlk 41717

Description: Properties of a pair of functions to be a simple walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.)

Hypotheses

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

Assertion

Ref	Expression
isupwlk	|- ( ( G e. W /\ F e. U /\ P e. Z ) -> ( F (UPWalks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )

Distinct variable groups:   k, G   k, F   P, k

Allowed substitution hints:   U( k)   I( k)   V( k)   W( k)   Z( k)

Proof of Theorem isupwlk

Dummy variables f p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 4654	. . 3 |- ( F (UPWalks ` G ) P <-> <. F , P >. e. (UPWalks ` G ) )
2		upwlksfval.v	. . . . . 6 |- V = (Vtx ` G )
3		upwlksfval.i	. . . . . 6 |- I = (iEdg ` G )
4	2, 3	upwlksfval 41716	. . . . 5 |- ( G e. W -> (UPWalks ` G ) = { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... ( # ` f ) ) --> V /\ A. k e. ( 0..^ ( # ` f ) ) ( I ` ( f ` k ) ) = { ( p ` k ) , ( p ` ( k + 1 ) ) } ) } )
5	4	3ad2ant1 1082	. . . 4 |- ( ( G e. W /\ F e. U /\ P e. Z ) -> (UPWalks ` G ) = { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... ( # ` f ) ) --> V /\ A. k e. ( 0..^ ( # ` f ) ) ( I ` ( f ` k ) ) = { ( p ` k ) , ( p ` ( k + 1 ) ) } ) } )
6	5	eleq2d 2687	. . 3 |- ( ( G e. W /\ F e. U /\ P e. Z ) -> ( <. F , P >. e. (UPWalks ` G ) <-> <. F , P >. e. { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... ( # ` f ) ) --> V /\ A. k e. ( 0..^ ( # ` f ) ) ( I ` ( f ` k ) ) = { ( p ` k ) , ( p ` ( k + 1 ) ) } ) } ) )
7	1, 6	syl5bb 272	. 2 |- ( ( G e. W /\ F e. U /\ P e. Z ) -> ( F (UPWalks ` G ) P <-> <. F , P >. e. { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... ( # ` f ) ) --> V /\ A. k e. ( 0..^ ( # ` f ) ) ( I ` ( f ` k ) ) = { ( p ` k ) , ( p ` ( k + 1 ) ) } ) } ) )
8		eleq1 2689	. . . . . 6 |- ( f = F -> ( f e. Word dom I <-> F e. Word dom I ) )
9	8	adantr 481	. . . . 5 |- ( ( f = F /\ p = P ) -> ( f e. Word dom I <-> F e. Word dom I ) )
10		simpr 477	. . . . . 6 |- ( ( f = F /\ p = P ) -> p = P )
11		fveq2 6191	. . . . . . . 8 |- ( f = F -> ( # ` f ) = ( # ` F ) )
12	11	oveq2d 6666	. . . . . . 7 |- ( f = F -> ( 0 ... ( # ` f ) ) = ( 0 ... ( # ` F ) ) )
13	12	adantr 481	. . . . . 6 |- ( ( f = F /\ p = P ) -> ( 0 ... ( # ` f ) ) = ( 0 ... ( # ` F ) ) )
14	10, 13	feq12d 6033	. . . . 5 |- ( ( f = F /\ p = P ) -> ( p : ( 0 ... ( # ` f ) ) --> V <-> P : ( 0 ... ( # ` F ) ) --> V ) )
15	11	oveq2d 6666	. . . . . . 7 |- ( f = F -> ( 0..^ ( # ` f ) ) = ( 0..^ ( # ` F ) ) )
16	15	adantr 481	. . . . . 6 |- ( ( f = F /\ p = P ) -> ( 0..^ ( # ` f ) ) = ( 0..^ ( # ` F ) ) )
17		fveq1 6190	. . . . . . . 8 |- ( f = F -> ( f ` k ) = ( F ` k ) )
18	17	fveq2d 6195	. . . . . . 7 |- ( f = F -> ( I ` ( f ` k ) ) = ( I ` ( F ` k ) ) )
19		fveq1 6190	. . . . . . . 8 |- ( p = P -> ( p ` k ) = ( P ` k ) )
20		fveq1 6190	. . . . . . . 8 |- ( p = P -> ( p ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) )
21	19, 20	preq12d 4276	. . . . . . 7 |- ( p = P -> { ( p ` k ) , ( p ` ( k + 1 ) ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
22	18, 21	eqeqan12d 2638	. . . . . 6 |- ( ( f = F /\ p = P ) -> ( ( I ` ( f ` k ) ) = { ( p ` k ) , ( p ` ( k + 1 ) ) } <-> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
23	16, 22	raleqbidv 3152	. . . . 5 |- ( ( f = F /\ p = P ) -> ( A. k e. ( 0..^ ( # ` f ) ) ( I ` ( f ` k ) ) = { ( p ` k ) , ( p ` ( k + 1 ) ) } <-> A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
24	9, 14, 23	3anbi123d 1399	. . . 4 |- ( ( f = F /\ p = P ) -> ( ( f e. Word dom I /\ p : ( 0 ... ( # ` f ) ) --> V /\ A. k e. ( 0..^ ( # ` f ) ) ( I ` ( f ` k ) ) = { ( p ` k ) , ( p ` ( k + 1 ) ) } ) <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
25	24	opelopabga 4988	. . 3 |- ( ( F e. U /\ P e. Z ) -> ( <. F , P >. e. { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... ( # ` f ) ) --> V /\ A. k e. ( 0..^ ( # ` f ) ) ( I ` ( f ` k ) ) = { ( p ` k ) , ( p ` ( k + 1 ) ) } ) } <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
26	25	3adant1 1079	. 2 |- ( ( G e. W /\ F e. U /\ P e. Z ) -> ( <. F , P >. e. { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... ( # ` f ) ) --> V /\ A. k e. ( 0..^ ( # ` f ) ) ( I ` ( f ` k ) ) = { ( p ` k ) , ( p ` ( k + 1 ) ) } ) } <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
27	7, 26	bitrd 268	1 |- ( ( G e. W /\ F e. U /\ P e. Z ) -> ( F (UPWalks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {cpr 4179   <.cop 4183   class class class wbr 4653   {copab 4712   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  UPWalkscupwlks 41714

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-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-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-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-upwlks 41715

This theorem is referenced by:  isupwlkg  41718  upwlkwlk  41720  upgrwlkupwlk  41721  upgrisupwlkALT  41723