Theorem iswlk 16247

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 4094	. . 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 16246	. . . . 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 1045	. . . 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 2301	. . 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 2294	. . . . . 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 5648	. . . . . . . 8 |- ( f = F -> (♯ ` f ) = (♯ ` F ) )
12	11	oveq2d 6044	. . . . . . 7 |- ( f = F -> ( 0 ... (♯ ` f ) ) = ( 0 ... (♯ ` F ) ) )
13	12	adantr 276	. . . . . 6 |- ( ( f = F /\ p = P ) -> ( 0 ... (♯ ` f ) ) = ( 0 ... (♯ ` F ) ) )
14	10, 13	feq12d 5479	. . . . 5 |- ( ( f = F /\ p = P ) -> ( p : ( 0 ... (♯ ` f ) ) --> V <-> P : ( 0 ... (♯ ` F ) ) --> V ) )
15	11	oveq2d 6044	. . . . . . 7 |- ( f = F -> ( 0..^ (♯ ` f ) ) = ( 0..^ (♯ ` F ) ) )
16	15	adantr 276	. . . . . 6 |- ( ( f = F /\ p = P ) -> ( 0..^ (♯ ` f ) ) = ( 0..^ (♯ ` F ) ) )
17		fveq1 5647	. . . . . . . . 9 |- ( p = P -> ( p ` k ) = ( P ` k ) )
18		fveq1 5647	. . . . . . . . 9 |- ( p = P -> ( p ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) )
19	17, 18	eqeq12d 2246	. . . . . . . 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 5647	. . . . . . . . 9 |- ( f = F -> ( f ` k ) = ( F ` k ) )
22	21	fveq2d 5652	. . . . . . . 8 |- ( f = F -> ( I ` ( f ` k ) ) = ( I ` ( F ` k ) ) )
23	17	sneqd 3686	. . . . . . . 8 |- ( p = P -> { ( p ` k ) } = { ( P ` k ) } )
24	22, 23	eqeqan12d 2247	. . . . . . 7 |- ( ( f = F /\ p = P ) -> ( ( I ` ( f ` k ) ) = { ( p ` k ) } <-> ( I ` ( F ` k ) ) = { ( P ` k ) } ) )
25	17, 18	preq12d 3760	. . . . . . . . 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 3259	. . . . . . 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 1001	. . . . . 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 2747	. . . . 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 1349	. . . 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 4363	. . 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 1042	. 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 986   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   {csn 3673   {cpr 3674   <.cop 3676   class class class wbr 4093   {copab 4154   dom cdm 4731   -->wf 5329   ` cfv 5333  (class class class)co 6028   0cc0 8075   1c1 8076   + caddc 8078   ...cfz 10288  ..^cfzo 10422  ♯chash 11083  Word cword 11162  Vtxcvtx 15936  iEdgciedg 15937  Walkscwlks 16241

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-z 9524  df-dec 9656  df-uz 9800  df-fz 10289  df-fzo 10423  df-ihash 11084  df-word 11163  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-wlks 16242

This theorem is referenced by:  wlkpropg  16248  iswlkg  16253  wlkvtxeledgg  16268