Theorem iswlk 16029

Description: Properties of a pair of functions to be/represent a walk. (Contributed by AV, 30-Dec-2020.)

Hypotheses

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

Assertion

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

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 iswlk

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

Step	Hyp	Ref	Expression
1		df-br 4083	. . 3 |- ( F (Walks ` G ) P <-> <. F , P >. e. (Walks ` G ) )
2		wksfval.v	. . . . . 6 |- V = (Vtx ` G )
3		wksfval.i	. . . . . 6 |- I = (iEdg ` G )
4	2, 3	wksfval 16028	. . . . 5 |- ( G e. W -> (Walks ` G ) = { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... (♯ ` f ) ) --> V /\ A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) ) } )
5	4	3ad2ant1 1042	. . . 4 |- ( ( G e. W /\ F e. U /\ P e. Z ) -> (Walks ` G ) = { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... (♯ ` f ) ) --> V /\ A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) ) } )
6	5	eleq2d 2299	. . 3 |- ( ( G e. W /\ F e. U /\ P e. Z ) -> ( <. F , P >. e. (Walks ` G ) <-> <. F , P >. e. { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... (♯ ` f ) ) --> V /\ A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) ) } ) )
7	1, 6	bitrid 192	. 2 |- ( ( G e. W /\ F e. U /\ P e. Z ) -> ( F (Walks ` G ) P <-> <. F , P >. e. { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... (♯ ` f ) ) --> V /\ A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) ) } ) )
8		eleq1 2292	. . . . . 6 |- ( f = F -> ( f e. Word dom I <-> F e. Word dom I ) )
9	8	adantr 276	. . . . 5 |- ( ( f = F /\ p = P ) -> ( f e. Word dom I <-> F e. Word dom I ) )
10		simpr 110	. . . . . 6 |- ( ( f = F /\ p = P ) -> p = P )
11		fveq2 5626	. . . . . . . 8 |- ( f = F -> (♯ ` f ) = (♯ ` F ) )
12	11	oveq2d 6016	. . . . . . 7 |- ( f = F -> ( 0 ... (♯ ` f ) ) = ( 0 ... (♯ ` F ) ) )
13	12	adantr 276	. . . . . 6 |- ( ( f = F /\ p = P ) -> ( 0 ... (♯ ` f ) ) = ( 0 ... (♯ ` F ) ) )
14	10, 13	feq12d 5462	. . . . 5 |- ( ( f = F /\ p = P ) -> ( p : ( 0 ... (♯ ` f ) ) --> V <-> P : ( 0 ... (♯ ` F ) ) --> V ) )
15	11	oveq2d 6016	. . . . . . 7 |- ( f = F -> ( 0..^ (♯ ` f ) ) = ( 0..^ (♯ ` F ) ) )
16	15	adantr 276	. . . . . 6 |- ( ( f = F /\ p = P ) -> ( 0..^ (♯ ` f ) ) = ( 0..^ (♯ ` F ) ) )
17		fveq1 5625	. . . . . . . . 9 |- ( p = P -> ( p ` k ) = ( P ` k ) )
18		fveq1 5625	. . . . . . . . 9 |- ( p = P -> ( p ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) )
19	17, 18	eqeq12d 2244	. . . . . . . 8 |- ( p = P -> ( ( p ` k ) = ( p ` ( k + 1 ) ) <-> ( P ` k ) = ( P ` ( k + 1 ) ) ) )
20	19	adantl 277	. . . . . . 7 |- ( ( f = F /\ p = P ) -> ( ( p ` k ) = ( p ` ( k + 1 ) ) <-> ( P ` k ) = ( P ` ( k + 1 ) ) ) )
21		fveq1 5625	. . . . . . . . 9 |- ( f = F -> ( f ` k ) = ( F ` k ) )
22	21	fveq2d 5630	. . . . . . . 8 |- ( f = F -> ( I ` ( f ` k ) ) = ( I ` ( F ` k ) ) )
23	17	sneqd 3679	. . . . . . . 8 |- ( p = P -> { ( p ` k ) } = { ( P ` k ) } )
24	22, 23	eqeqan12d 2245	. . . . . . 7 |- ( ( f = F /\ p = P ) -> ( ( I ` ( f ` k ) ) = { ( p ` k ) } <-> ( I ` ( F ` k ) ) = { ( P ` k ) } ) )
25	17, 18	preq12d 3751	. . . . . . . . 9 |- ( p = P -> { ( p ` k ) , ( p ` ( k + 1 ) ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
26	25	adantl 277	. . . . . . . 8 |- ( ( f = F /\ p = P ) -> { ( p ` k ) , ( p ` ( k + 1 ) ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
27	22	adantr 276	. . . . . . . 8 |- ( ( f = F /\ p = P ) -> ( I ` ( f ` k ) ) = ( I ` ( F ` k ) ) )
28	26, 27	sseq12d 3255	. . . . . . 7 |- ( ( f = F /\ p = P ) -> ( { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) <-> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
29	20, 24, 28	ifpbi123d 998	. . . . . 6 |- ( ( f = F /\ p = P ) -> (if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) <-> if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
30	16, 29	raleqbidv 2744	. . . . 5 |- ( ( f = F /\ p = P ) -> ( A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) <-> A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
31	9, 14, 30	3anbi123d 1346	. . . 4 |- ( ( f = F /\ p = P ) -> ( ( f e. Word dom I /\ p : ( 0 ... (♯ ` f ) ) --> V /\ A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) ) <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
32	31	opelopabga 4350	. . 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 ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) ) } <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
33	32	3adant1 1039	. 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 ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) ) } <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
34	7, 33	bitrd 188	1 |- ( ( G e. W /\ F e. U /\ P e. Z ) -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  if-wif 983   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   {csn 3666   {cpr 3667   <.cop 3669   class class class wbr 4082   {copab 4143   dom cdm 4718   -->wf 5313   ` cfv 5317  (class class class)co 6000   0cc0 7995   1c1 7996   + caddc 7998   ...cfz 10200  ..^cfzo 10334  ♯chash 10992  Word cword 11066  Vtxcvtx 15807  iEdgciedg 15808  Walkscwlks 16024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-er 6678  df-map 6795  df-en 6886  df-dom 6887  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-9 9172  df-n0 9366  df-z 9443  df-dec 9575  df-uz 9719  df-fz 10201  df-fzo 10335  df-ihash 10993  df-word 11067  df-ndx 13030  df-slot 13031  df-base 13033  df-edgf 15800  df-vtx 15809  df-iedg 15810  df-wlks 16025

This theorem is referenced by:  wlkpropg  16030  iswlkg  16032  wlkvtxeledgg  16041