Theorem wksfval 16063

Description: The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.)

Hypotheses

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

Assertion

Ref	Expression
wksfval	|- ( 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 ) ) ) ) } )

Distinct variable groups:   f, G, k, p   f, I, p   V, p   f, W

Allowed substitution hints:   I( k)   V( f, k)   W( k, p)

Proof of Theorem wksfval

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wlks 16059	. 2 |- Walks = ( g e. _V |-> { <. f , p >. | ( f e. Word dom (iEdg ` g ) /\ p : ( 0 ... (♯ ` f ) ) --> (Vtx ` g ) /\ A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( (iEdg ` g ) ` ( f ` k ) ) ) ) } )
2		fveq2 5629	. . . . . . . 8 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
3		wksfval.i	. . . . . . . 8 |- I = (iEdg ` G )
4	2, 3	eqtr4di 2280	. . . . . . 7 |- ( g = G -> (iEdg ` g ) = I )
5	4	dmeqd 4925	. . . . . 6 |- ( g = G -> dom (iEdg ` g ) = dom I )
6		wrdeq 11106	. . . . . 6 |- ( dom (iEdg ` g ) = dom I -> Word dom (iEdg ` g ) = Word dom I )
7	5, 6	syl 14	. . . . 5 |- ( g = G -> Word dom (iEdg ` g ) = Word dom I )
8	7	eleq2d 2299	. . . 4 |- ( g = G -> ( f e. Word dom (iEdg ` g ) <-> f e. Word dom I ) )
9		fveq2 5629	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
10		wksfval.v	. . . . . 6 |- V = (Vtx ` G )
11	9, 10	eqtr4di 2280	. . . . 5 |- ( g = G -> (Vtx ` g ) = V )
12	11	feq3d 5462	. . . 4 |- ( g = G -> ( p : ( 0 ... (♯ ` f ) ) --> (Vtx ` g ) <-> p : ( 0 ... (♯ ` f ) ) --> V ) )
13	4	fveq1d 5631	. . . . . . 7 |- ( g = G -> ( (iEdg ` g ) ` ( f ` k ) ) = ( I ` ( f ` k ) ) )
14	13	eqeq1d 2238	. . . . . 6 |- ( g = G -> ( ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } <-> ( I ` ( f ` k ) ) = { ( p ` k ) } ) )
15	13	sseq2d 3254	. . . . . 6 |- ( g = G -> ( { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( (iEdg ` g ) ` ( f ` k ) ) <-> { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) )
16	14, 15	ifpbi23d 999	. . . . 5 |- ( g = G -> (if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( (iEdg ` g ) ` ( f ` k ) ) ) <-> if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) ) )
17	16	ralbidv 2530	. . . 4 |- ( g = G -> ( A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( (iEdg ` g ) ` ( 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 ) ) ) ) )
18	8, 12, 17	3anbi123d 1346	. . 3 |- ( g = G -> ( ( f e. Word dom (iEdg ` g ) /\ p : ( 0 ... (♯ ` f ) ) --> (Vtx ` g ) /\ A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( (iEdg ` g ) ` ( 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 ) ) ) ) ) )
19	18	opabbidv 4150	. 2 |- ( g = G -> { <. f , p >. | ( f e. Word dom (iEdg ` g ) /\ p : ( 0 ... (♯ ` f ) ) --> (Vtx ` g ) /\ A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( (iEdg ` g ) ` ( f ` k ) ) ) ) } = { <. 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 ) ) ) ) } )
20		elex 2811	. 2 |- ( G e. W -> G e. _V )
21		3anass 1006	. . . 4 |- ( ( 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 ) ) ) ) ) )
22	21	opabbii 4151	. . 3 |- { <. 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 , 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 ) ) ) ) ) }
23		iedgex 15835	. . . . . . 7 |- ( G e. W -> (iEdg ` G ) e. _V )
24	3, 23	eqeltrid 2316	. . . . . 6 |- ( G e. W -> I e. _V )
25	24	dmexd 4990	. . . . 5 |- ( G e. W -> dom I e. _V )
26		wrdexg 11095	. . . . 5 |- ( dom I e. _V -> Word dom I e. _V )
27	25, 26	syl 14	. . . 4 |- ( G e. W -> Word dom I e. _V )
28		0zd 9469	. . . . . . 7 |- ( f e. Word dom I -> 0 e. ZZ )
29		lencl 11088	. . . . . . . 8 |- ( f e. Word dom I -> (♯ ` f ) e. NN0 )
30	29	nn0zd 9578	. . . . . . 7 |- ( f e. Word dom I -> (♯ ` f ) e. ZZ )
31	28, 30	fzfigd 10665	. . . . . 6 |- ( f e. Word dom I -> ( 0 ... (♯ ` f ) ) e. Fin )
32		vtxex 15834	. . . . . . 7 |- ( G e. W -> (Vtx ` G ) e. _V )
33	10, 32	eqeltrid 2316	. . . . . 6 |- ( G e. W -> V e. _V )
34		mapex 6809	. . . . . 6 |- ( ( ( 0 ... (♯ ` f ) ) e. Fin /\ V e. _V ) -> { p | p : ( 0 ... (♯ ` f ) ) --> V } e. _V )
35	31, 33, 34	syl2anr 290	. . . . 5 |- ( ( G e. W /\ f e. Word dom I ) -> { p | p : ( 0 ... (♯ ` f ) ) --> V } e. _V )
36		simpl 109	. . . . . . 7 |- ( ( 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 ) ) ) ) -> p : ( 0 ... (♯ ` f ) ) --> V )
37	36	ss2abi 3296	. . . . . 6 |- { p | ( 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 ) ) ) ) } C_ { p | p : ( 0 ... (♯ ` f ) ) --> V }
38	37	a1i 9	. . . . 5 |- ( ( G e. W /\ f e. Word dom I ) -> { p | ( 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 ) ) ) ) } C_ { p | p : ( 0 ... (♯ ` f ) ) --> V } )
39	35, 38	ssexd 4224	. . . 4 |- ( ( G e. W /\ f e. Word dom I ) -> { p | ( 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 ) ) ) ) } e. _V )
40	27, 39	opabex3d 6272	. . 3 |- ( G e. W -> { <. 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 ) ) ) ) ) } e. _V )
41	22, 40	eqeltrid 2316	. 2 |- ( G e. W -> { <. 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 ) ) ) ) } e. _V )
42	1, 19, 20, 41	fvmptd3 5730	1 |- ( 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 ) ) ) ) } )

Syntax hints:   -> wi 4   /\ wa 104  if-wif 983   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   _Vcvv 2799   C_ wss 3197   {csn 3666   {cpr 3667   {copab 4144   dom cdm 4719   -->wf 5314   ` cfv 5318  (class class class)co 6007   Fincfn 6895   0cc0 8010   1c1 8011   + caddc 8013   ...cfz 10216  ..^cfzo 10350  ♯chash 11009  Word cword 11084  Vtxcvtx 15828  iEdgciedg 15829  Walkscwlks 16058

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126

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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-map 6805  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-z 9458  df-dec 9590  df-uz 9734  df-fz 10217  df-fzo 10351  df-ihash 11010  df-word 11085  df-ndx 13050  df-slot 13051  df-base 13053  df-edgf 15821  df-vtx 15830  df-iedg 15831  df-wlks 16059

This theorem is referenced by:  iswlk  16064  wlkpropg  16065  wlkex  16066  wlkv  16067  wlkvg  16069