Theorem wksfval 16443

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 16439	. 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 5675	. . . . . . . 8 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
3		wksfval.i	. . . . . . . 8 |- I = (iEdg ` G )
4	2, 3	eqtr4di 2285	. . . . . . 7 |- ( g = G -> (iEdg ` g ) = I )
5	4	dmeqd 4963	. . . . . 6 |- ( g = G -> dom (iEdg ` g ) = dom I )
6		wrdeq 11271	. . . . . 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 2304	. . . 4 |- ( g = G -> ( f e. Word dom (iEdg ` g ) <-> f e. Word dom I ) )
9		fveq2 5675	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
10		wksfval.v	. . . . . 6 |- V = (Vtx ` G )
11	9, 10	eqtr4di 2285	. . . . 5 |- ( g = G -> (Vtx ` g ) = V )
12	11	feq3d 5502	. . . 4 |- ( g = G -> ( p : ( 0 ... (♯ ` f ) ) --> (Vtx ` g ) <-> p : ( 0 ... (♯ ` f ) ) --> V ) )
13	4	fveq1d 5677	. . . . . . 7 |- ( g = G -> ( (iEdg ` g ) ` ( f ` k ) ) = ( I ` ( f ` k ) ) )
14	13	eqeq1d 2243	. . . . . 6 |- ( g = G -> ( ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } <-> ( I ` ( f ` k ) ) = { ( p ` k ) } ) )
15	13	sseq2d 3272	. . . . . 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 1002	. . . . 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 2544	. . . 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 1349	. . 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 4181	. 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 2827	. 2 |- ( G e. W -> G e. _V )
21		3anass 1009	. . . 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 4182	. . 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 16140	. . . . . . 7 |- ( G e. W -> (iEdg ` G ) e. _V )
24	3, 23	eqeltrid 2321	. . . . . 6 |- ( G e. W -> I e. _V )
25	24	dmexd 5028	. . . . 5 |- ( G e. W -> dom I e. _V )
26		wrdexg 11260	. . . . 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 9606	. . . . . . 7 |- ( f e. Word dom I -> 0 e. ZZ )
29		lencl 11253	. . . . . . . 8 |- ( f e. Word dom I -> (♯ ` f ) e. NN0 )
30	29	nn0zd 9716	. . . . . . 7 |- ( f e. Word dom I -> (♯ ` f ) e. ZZ )
31	28, 30	fzfigd 10817	. . . . . 6 |- ( f e. Word dom I -> ( 0 ... (♯ ` f ) ) e. Fin )
32		vtxex 16139	. . . . . . 7 |- ( G e. W -> (Vtx ` G ) e. _V )
33	10, 32	eqeltrid 2321	. . . . . 6 |- ( G e. W -> V e. _V )
34		mapex 6901	. . . . . 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 3314	. . . . . 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 4255	. . . 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 6323	. . 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 2321	. 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 5776	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 986   /\ w3a 1005   = wceq 1398   e. wcel 2205   {cab 2220   A.wral 2522   _Vcvv 2815   C_ wss 3214   {csn 3694   {cpr 3695   {copab 4175   dom cdm 4754   -->wf 5353   ` cfv 5357  (class class class)co 6058   Fincfn 6988   0cc0 8143   1c1 8144   + caddc 8146   ...cfz 10361  ..^cfzo 10498  ♯chash 11163  Word cword 11249  Vtxcvtx 16133  iEdgciedg 16134  Walkscwlks 16438

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-wlks 16439

This theorem is referenced by:  iswlk  16444  wlkpropg  16445  wlkex  16446  wlkv  16447  wlkvg  16449