Theorem wksfval 16172

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 16168	. 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 5639	. . . . . . . 8 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
3		wksfval.i	. . . . . . . 8 |- I = (iEdg ` G )
4	2, 3	eqtr4di 2282	. . . . . . 7 |- ( g = G -> (iEdg ` g ) = I )
5	4	dmeqd 4933	. . . . . 6 |- ( g = G -> dom (iEdg ` g ) = dom I )
6		wrdeq 11134	. . . . . 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 2301	. . . 4 |- ( g = G -> ( f e. Word dom (iEdg ` g ) <-> f e. Word dom I ) )
9		fveq2 5639	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
10		wksfval.v	. . . . . 6 |- V = (Vtx ` G )
11	9, 10	eqtr4di 2282	. . . . 5 |- ( g = G -> (Vtx ` g ) = V )
12	11	feq3d 5471	. . . 4 |- ( g = G -> ( p : ( 0 ... (♯ ` f ) ) --> (Vtx ` g ) <-> p : ( 0 ... (♯ ` f ) ) --> V ) )
13	4	fveq1d 5641	. . . . . . 7 |- ( g = G -> ( (iEdg ` g ) ` ( f ` k ) ) = ( I ` ( f ` k ) ) )
14	13	eqeq1d 2240	. . . . . 6 |- ( g = G -> ( ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } <-> ( I ` ( f ` k ) ) = { ( p ` k ) } ) )
15	13	sseq2d 3257	. . . . . 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 1001	. . . . 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 2532	. . . 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 1348	. . 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 4155	. 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 2814	. 2 |- ( G e. W -> G e. _V )
21		3anass 1008	. . . 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 4156	. . 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 15869	. . . . . . 7 |- ( G e. W -> (iEdg ` G ) e. _V )
24	3, 23	eqeltrid 2318	. . . . . 6 |- ( G e. W -> I e. _V )
25	24	dmexd 4998	. . . . 5 |- ( G e. W -> dom I e. _V )
26		wrdexg 11123	. . . . 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 9490	. . . . . . 7 |- ( f e. Word dom I -> 0 e. ZZ )
29		lencl 11116	. . . . . . . 8 |- ( f e. Word dom I -> (♯ ` f ) e. NN0 )
30	29	nn0zd 9599	. . . . . . 7 |- ( f e. Word dom I -> (♯ ` f ) e. ZZ )
31	28, 30	fzfigd 10692	. . . . . 6 |- ( f e. Word dom I -> ( 0 ... (♯ ` f ) ) e. Fin )
32		vtxex 15868	. . . . . . 7 |- ( G e. W -> (Vtx ` G ) e. _V )
33	10, 32	eqeltrid 2318	. . . . . 6 |- ( G e. W -> V e. _V )
34		mapex 6822	. . . . . 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 3299	. . . . . 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 4229	. . . 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 6282	. . 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 2318	. 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 5740	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 985   /\ w3a 1004   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   _Vcvv 2802   C_ wss 3200   {csn 3669   {cpr 3670   {copab 4149   dom cdm 4725   -->wf 5322   ` cfv 5326  (class class class)co 6017   Fincfn 6908   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:  iswlk  16173  wlkpropg  16174  wlkex  16175  wlkv  16176  wlkvg  16178