Theorem wksfval 16477

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 16473	. 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 5690	. . . . . . . 8 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
3		wksfval.i	. . . . . . . 8 |- I = (iEdg ` G )
4	2, 3	eqtr4di 2289	. . . . . . 7 |- ( g = G -> (iEdg ` g ) = I )
5	4	dmeqd 4978	. . . . . 6 |- ( g = G -> dom (iEdg ` g ) = dom I )
6		wrdeq 11304	. . . . . 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 2308	. . . 4 |- ( g = G -> ( f e. Word dom (iEdg ` g ) <-> f e. Word dom I ) )
9		fveq2 5690	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
10		wksfval.v	. . . . . 6 |- V = (Vtx ` G )
11	9, 10	eqtr4di 2289	. . . . 5 |- ( g = G -> (Vtx ` g ) = V )
12	11	feq3d 5517	. . . 4 |- ( g = G -> ( p : ( 0 ... (♯ ` f ) ) --> (Vtx ` g ) <-> p : ( 0 ... (♯ ` f ) ) --> V ) )
13	4	fveq1d 5692	. . . . . . 7 |- ( g = G -> ( (iEdg ` g ) ` ( f ` k ) ) = ( I ` ( f ` k ) ) )
14	13	eqeq1d 2247	. . . . . 6 |- ( g = G -> ( ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } <-> ( I ` ( f ` k ) ) = { ( p ` k ) } ) )
15	13	sseq2d 3278	. . . . . 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 1006	. . . . 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 2550	. . . 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 1353	. . 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 4192	. 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 2833	. 2 |- ( G e. W -> G e. _V )
21		3anass 1013	. . . 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 4193	. . 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 16174	. . . . . . 7 |- ( G e. W -> (iEdg ` G ) e. _V )
24	3, 23	eqeltrid 2325	. . . . . 6 |- ( G e. W -> I e. _V )
25	24	dmexd 5043	. . . . 5 |- ( G e. W -> dom I e. _V )
26		wrdexg 11293	. . . . 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 9635	. . . . . . 7 |- ( f e. Word dom I -> 0 e. ZZ )
29		lencl 11286	. . . . . . . 8 |- ( f e. Word dom I -> (♯ ` f ) e. NN0 )
30	29	nn0zd 9745	. . . . . . 7 |- ( f e. Word dom I -> (♯ ` f ) e. ZZ )
31	28, 30	fzfigd 10846	. . . . . 6 |- ( f e. Word dom I -> ( 0 ... (♯ ` f ) ) e. Fin )
32		vtxex 16173	. . . . . . 7 |- ( G e. W -> (Vtx ` G ) e. _V )
33	10, 32	eqeltrid 2325	. . . . . 6 |- ( G e. W -> V e. _V )
34		mapex 6918	. . . . . 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 3320	. . . . . 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 4268	. . . 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 6340	. . 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 2325	. 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 5793	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 990   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   _Vcvv 2821   C_ wss 3220   {csn 3705   {cpr 3706   {copab 4186   dom cdm 4769   -->wf 5368   ` cfv 5372  (class class class)co 6075   Fincfn 7012   0cc0 8169   1c1 8170   + caddc 8172   ...cfz 10390  ..^cfzo 10527  ♯chash 11192  Word cword 11282  Vtxcvtx 16167  iEdgciedg 16168  Walkscwlks 16472

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-map 6914  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-z 9624  df-dec 9757  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-wlks 16473

This theorem is referenced by:  iswlk  16478  wlkpropg  16479  wlkex  16480  wlkv  16481  wlkvg  16483