Theorem iswlk 16173

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 4089	. . 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 16172	. . . . 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 1044	. . . 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 5639	. . . . . . . 8 |- ( f = F -> (♯ ` f ) = (♯ ` F ) )
12	11	oveq2d 6033	. . . . . . 7 |- ( f = F -> ( 0 ... (♯ ` f ) ) = ( 0 ... (♯ ` F ) ) )
13	12	adantr 276	. . . . . 6 |- ( ( f = F /\ p = P ) -> ( 0 ... (♯ ` f ) ) = ( 0 ... (♯ ` F ) ) )
14	10, 13	feq12d 5472	. . . . 5 |- ( ( f = F /\ p = P ) -> ( p : ( 0 ... (♯ ` f ) ) --> V <-> P : ( 0 ... (♯ ` F ) ) --> V ) )
15	11	oveq2d 6033	. . . . . . 7 |- ( f = F -> ( 0..^ (♯ ` f ) ) = ( 0..^ (♯ ` F ) ) )
16	15	adantr 276	. . . . . 6 |- ( ( f = F /\ p = P ) -> ( 0..^ (♯ ` f ) ) = ( 0..^ (♯ ` F ) ) )
17		fveq1 5638	. . . . . . . . 9 |- ( p = P -> ( p ` k ) = ( P ` k ) )
18		fveq1 5638	. . . . . . . . 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 5638	. . . . . . . . 9 |- ( f = F -> ( f ` k ) = ( F ` k ) )
22	21	fveq2d 5643	. . . . . . . 8 |- ( f = F -> ( I ` ( f ` k ) ) = ( I ` ( F ` k ) ) )
23	17	sneqd 3682	. . . . . . . 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 3756	. . . . . . . . 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 3258	. . . . . . 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 1000	. . . . . 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 2746	. . . . 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 1348	. . . 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 4357	. . 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 1041	. 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 985   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   {csn 3669   {cpr 3670   <.cop 3672   class class class wbr 4088   {copab 4149   dom cdm 4725   -->wf 5322   ` cfv 5326  (class class class)co 6017   0cc0 8031   1c1 8032   + caddc 8034   ...cfz 10242  ..^cfzo 10376  ♯chash 11036  Word cword 11112  Vtxcvtx 15862  iEdgciedg 15863  Walkscwlks 16167

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-ifp 986  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-map 6818  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-z 9479  df-dec 9611  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-wlks 16168

This theorem is referenced by:  wlkpropg  16174  iswlkg  16179  wlkvtxeledgg  16194