Theorem wksfval 26505

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 26495	. . 3 |- 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	1	a1i 11	. 2 |- ( G e. W -> 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 ) ) ) ) } ) )
3		fveq2 6191	. . . . . . . . 9 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
4		wksfval.i	. . . . . . . . 9 |- I = (iEdg ` G )
5	3, 4	syl6eqr 2674	. . . . . . . 8 |- ( g = G -> (iEdg ` g ) = I )
6	5	dmeqd 5326	. . . . . . 7 |- ( g = G -> dom (iEdg ` g ) = dom I )
7		wrdeq 13327	. . . . . . 7 |- ( dom (iEdg ` g ) = dom I -> Word dom (iEdg ` g ) = Word dom I )
8	6, 7	syl 17	. . . . . 6 |- ( g = G -> Word dom (iEdg ` g ) = Word dom I )
9	8	eleq2d 2687	. . . . 5 |- ( g = G -> ( f e. Word dom (iEdg ` g ) <-> f e. Word dom I ) )
10		fveq2 6191	. . . . . . 7 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
11		wksfval.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	syl6eqr 2674	. . . . . 6 |- ( g = G -> (Vtx ` g ) = V )
13	12	feq3d 6032	. . . . 5 |- ( g = G -> ( p : ( 0 ... ( # ` f ) ) --> (Vtx ` g ) <-> p : ( 0 ... ( # ` f ) ) --> V ) )
14		biidd 252	. . . . . . 7 |- ( g = G -> ( ( p ` k ) = ( p ` ( k + 1 ) ) <-> ( p ` k ) = ( p ` ( k + 1 ) ) ) )
15	5	fveq1d 6193	. . . . . . . 8 |- ( g = G -> ( (iEdg ` g ) ` ( f ` k ) ) = ( I ` ( f ` k ) ) )
16	15	eqeq1d 2624	. . . . . . 7 |- ( g = G -> ( ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } <-> ( I ` ( f ` k ) ) = { ( p ` k ) } ) )
17	15	sseq2d 3633	. . . . . . 7 |- ( g = G -> ( { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( (iEdg ` g ) ` ( f ` k ) ) <-> { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) )
18	14, 16, 17	ifpbi123d 1027	. . . . . 6 |- ( 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 ) ) ) ) )
19	18	ralbidv 2986	. . . . 5 |- ( 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 ) ) ) ) )
20	9, 13, 19	3anbi123d 1399	. . . 4 |- ( 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 ) ) ) ) ) )
21	20	opabbidv 4716	. . 3 |- ( 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 ) ) ) ) } )
22	21	adantl 482	. 2 |- ( ( G e. W /\ 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 ) ) ) ) } )
23		elex 3212	. 2 |- ( G e. W -> G e. _V )
24		3anass 1042	. . . 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 ) ) ) ) ) )
25	24	opabbii 4717	. . 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 ) ) ) ) ) }
26		fvex 6201	. . . . . . 7 |- (iEdg ` G ) e. _V
27	4, 26	eqeltri 2697	. . . . . 6 |- I e. _V
28	27	dmex 7099	. . . . 5 |- dom I e. _V
29		wrdexg 13315	. . . . 5 |- ( dom I e. _V -> Word dom I e. _V )
30	28, 29	mp1i 13	. . . 4 |- ( G e. W -> Word dom I e. _V )
31		ovex 6678	. . . . . 6 |- ( 0 ... ( # ` f ) ) e. _V
32		fvex 6201	. . . . . . . 8 |- (Vtx ` G ) e. _V
33	11, 32	eqeltri 2697	. . . . . . 7 |- V e. _V
34	33	a1i 11	. . . . . 6 |- ( ( G e. W /\ f e. Word dom I ) -> V e. _V )
35		mapex 7863	. . . . . 6 |- ( ( ( 0 ... ( # ` f ) ) e. _V /\ V e. _V ) -> { p | p : ( 0 ... ( # ` f ) ) --> V } e. _V )
36	31, 34, 35	sylancr 695	. . . . 5 |- ( ( G e. W /\ f e. Word dom I ) -> { p | p : ( 0 ... ( # ` f ) ) --> V } e. _V )
37		simpl 473	. . . . . . 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 )
38	37	ss2abi 3674	. . . . . 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 }
39	38	a1i 11	. . . . 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 } )
40	36, 39	ssexd 4805	. . . 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 )
41	30, 40	opabex3d 7145	. . 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 )
42	25, 41	syl5eqel 2705	. 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 )
43	2, 22, 23, 42	fvmptd 6288	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 384  if-wif 1012   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   C_ wss 3574   {csn 4177   {cpr 4179   {copab 4712   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Walkscwlks 26492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495

This theorem is referenced by:  iswlk  26506  wlkprop  26507  wlkv  26508