Theorem ewlksfval 26497

Description: The set of s-walks of edges (in a hypergraph). (Contributed by AV, 4-Jan-2021.)

Hypothesis

Ref	Expression
ewlksfval.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
ewlksfval	|- ( ( G e. W /\ S e. NN0* ) -> ( G EdgWalks S ) = { f | ( f e. Word dom I /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( I ` ( f ` ( k - 1 ) ) ) i^i ( I ` ( f ` k ) ) ) ) ) } )

Distinct variable groups:   f, G, k   S, f, k   f, W, k

Allowed substitution hints:   I( f, k)

Proof of Theorem ewlksfval

Dummy variables g i s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ewlks 26494	. . . 4 |- EdgWalks = ( g e. _V , s e. NN0* |-> { f | [. (iEdg ` g ) / i ]. ( f e. Word dom i /\ A. k e. ( 1..^ ( # ` f ) ) s <_ ( # ` ( ( i ` ( f ` ( k - 1 ) ) ) i^i ( i ` ( f ` k ) ) ) ) ) } )
2	1	a1i 11	. . 3 |- ( ( G e. W /\ S e. NN0* ) -> EdgWalks = ( g e. _V , s e. NN0* |-> { f | [. (iEdg ` g ) / i ]. ( f e. Word dom i /\ A. k e. ( 1..^ ( # ` f ) ) s <_ ( # ` ( ( i ` ( f ` ( k - 1 ) ) ) i^i ( i ` ( f ` k ) ) ) ) ) } ) )
3		fvexd 6203	. . . . . 6 |- ( ( g = G /\ s = S ) -> (iEdg ` g ) e. _V )
4		simpr 477	. . . . . . . . . . 11 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> i = (iEdg ` g ) )
5		fveq2 6191	. . . . . . . . . . . . 13 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
6	5	adantr 481	. . . . . . . . . . . 12 |- ( ( g = G /\ s = S ) -> (iEdg ` g ) = (iEdg ` G ) )
7	6	adantr 481	. . . . . . . . . . 11 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> (iEdg ` g ) = (iEdg ` G ) )
8	4, 7	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> i = (iEdg ` G ) )
9	8	dmeqd 5326	. . . . . . . . 9 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> dom i = dom (iEdg ` G ) )
10		wrdeq 13327	. . . . . . . . 9 |- ( dom i = dom (iEdg ` G ) -> Word dom i = Word dom (iEdg ` G ) )
11	9, 10	syl 17	. . . . . . . 8 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> Word dom i = Word dom (iEdg ` G ) )
12	11	eleq2d 2687	. . . . . . 7 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> ( f e. Word dom i <-> f e. Word dom (iEdg ` G ) ) )
13		simpr 477	. . . . . . . . . 10 |- ( ( g = G /\ s = S ) -> s = S )
14	13	adantr 481	. . . . . . . . 9 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> s = S )
15	8	fveq1d 6193	. . . . . . . . . . 11 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> ( i ` ( f ` ( k - 1 ) ) ) = ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) )
16	8	fveq1d 6193	. . . . . . . . . . 11 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> ( i ` ( f ` k ) ) = ( (iEdg ` G ) ` ( f ` k ) ) )
17	15, 16	ineq12d 3815	. . . . . . . . . 10 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> ( ( i ` ( f ` ( k - 1 ) ) ) i^i ( i ` ( f ` k ) ) ) = ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) )
18	17	fveq2d 6195	. . . . . . . . 9 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> ( # ` ( ( i ` ( f ` ( k - 1 ) ) ) i^i ( i ` ( f ` k ) ) ) ) = ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) )
19	14, 18	breq12d 4666	. . . . . . . 8 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> ( s <_ ( # ` ( ( i ` ( f ` ( k - 1 ) ) ) i^i ( i ` ( f ` k ) ) ) ) <-> S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) ) )
20	19	ralbidv 2986	. . . . . . 7 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> ( A. k e. ( 1..^ ( # ` f ) ) s <_ ( # ` ( ( i ` ( f ` ( k - 1 ) ) ) i^i ( i ` ( f ` k ) ) ) ) <-> A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) ) )
21	12, 20	anbi12d 747	. . . . . 6 |- ( ( ( g = G /\ s = S ) /\ i = (iEdg ` g ) ) -> ( ( f e. Word dom i /\ A. k e. ( 1..^ ( # ` f ) ) s <_ ( # ` ( ( i ` ( f ` ( k - 1 ) ) ) i^i ( i ` ( f ` k ) ) ) ) ) <-> ( f e. Word dom (iEdg ` G ) /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) ) ) )
22	3, 21	sbcied 3472	. . . . 5 |- ( ( g = G /\ s = S ) -> ( [. (iEdg ` g ) / i ]. ( f e. Word dom i /\ A. k e. ( 1..^ ( # ` f ) ) s <_ ( # ` ( ( i ` ( f ` ( k - 1 ) ) ) i^i ( i ` ( f ` k ) ) ) ) ) <-> ( f e. Word dom (iEdg ` G ) /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) ) ) )
23	22	abbidv 2741	. . . 4 |- ( ( g = G /\ s = S ) -> { f | [. (iEdg ` g ) / i ]. ( f e. Word dom i /\ A. k e. ( 1..^ ( # ` f ) ) s <_ ( # ` ( ( i ` ( f ` ( k - 1 ) ) ) i^i ( i ` ( f ` k ) ) ) ) ) } = { f | ( f e. Word dom (iEdg ` G ) /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) ) } )
24	23	adantl 482	. . 3 |- ( ( ( G e. W /\ S e. NN0* ) /\ ( g = G /\ s = S ) ) -> { f | [. (iEdg ` g ) / i ]. ( f e. Word dom i /\ A. k e. ( 1..^ ( # ` f ) ) s <_ ( # ` ( ( i ` ( f ` ( k - 1 ) ) ) i^i ( i ` ( f ` k ) ) ) ) ) } = { f | ( f e. Word dom (iEdg ` G ) /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) ) } )
25		elex 3212	. . . 4 |- ( G e. W -> G e. _V )
26	25	adantr 481	. . 3 |- ( ( G e. W /\ S e. NN0* ) -> G e. _V )
27		simpr 477	. . 3 |- ( ( G e. W /\ S e. NN0* ) -> S e. NN0* )
28		df-rab 2921	. . . 4 |- { f e. Word dom (iEdg ` G ) | A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) } = { f | ( f e. Word dom (iEdg ` G ) /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) ) }
29		fvex 6201	. . . . . . . 8 |- (iEdg ` G ) e. _V
30	29	dmex 7099	. . . . . . 7 |- dom (iEdg ` G ) e. _V
31	30	wrdexi 13317	. . . . . 6 |- Word dom (iEdg ` G ) e. _V
32	31	rabex 4813	. . . . 5 |- { f e. Word dom (iEdg ` G ) | A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) } e. _V
33	32	a1i 11	. . . 4 |- ( ( G e. W /\ S e. NN0* ) -> { f e. Word dom (iEdg ` G ) | A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) } e. _V )
34	28, 33	syl5eqelr 2706	. . 3 |- ( ( G e. W /\ S e. NN0* ) -> { f | ( f e. Word dom (iEdg ` G ) /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) ) } e. _V )
35	2, 24, 26, 27, 34	ovmpt2d 6788	. 2 |- ( ( G e. W /\ S e. NN0* ) -> ( G EdgWalks S ) = { f | ( f e. Word dom (iEdg ` G ) /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) ) } )
36		ewlksfval.i	. . . . . . . . 9 |- I = (iEdg ` G )
37	36	eqcomi 2631	. . . . . . . 8 |- (iEdg ` G ) = I
38	37	a1i 11	. . . . . . 7 |- ( ( G e. W /\ S e. NN0* ) -> (iEdg ` G ) = I )
39	38	dmeqd 5326	. . . . . 6 |- ( ( G e. W /\ S e. NN0* ) -> dom (iEdg ` G ) = dom I )
40		wrdeq 13327	. . . . . 6 |- ( dom (iEdg ` G ) = dom I -> Word dom (iEdg ` G ) = Word dom I )
41	39, 40	syl 17	. . . . 5 |- ( ( G e. W /\ S e. NN0* ) -> Word dom (iEdg ` G ) = Word dom I )
42	41	eleq2d 2687	. . . 4 |- ( ( G e. W /\ S e. NN0* ) -> ( f e. Word dom (iEdg ` G ) <-> f e. Word dom I ) )
43	38	fveq1d 6193	. . . . . . . 8 |- ( ( G e. W /\ S e. NN0* ) -> ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) = ( I ` ( f ` ( k - 1 ) ) ) )
44	38	fveq1d 6193	. . . . . . . 8 |- ( ( G e. W /\ S e. NN0* ) -> ( (iEdg ` G ) ` ( f ` k ) ) = ( I ` ( f ` k ) ) )
45	43, 44	ineq12d 3815	. . . . . . 7 |- ( ( G e. W /\ S e. NN0* ) -> ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) = ( ( I ` ( f ` ( k - 1 ) ) ) i^i ( I ` ( f ` k ) ) ) )
46	45	fveq2d 6195	. . . . . 6 |- ( ( G e. W /\ S e. NN0* ) -> ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) = ( # ` ( ( I ` ( f ` ( k - 1 ) ) ) i^i ( I ` ( f ` k ) ) ) ) )
47	46	breq2d 4665	. . . . 5 |- ( ( G e. W /\ S e. NN0* ) -> ( S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) <-> S <_ ( # ` ( ( I ` ( f ` ( k - 1 ) ) ) i^i ( I ` ( f ` k ) ) ) ) ) )
48	47	ralbidv 2986	. . . 4 |- ( ( G e. W /\ S e. NN0* ) -> ( A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) <-> A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( I ` ( f ` ( k - 1 ) ) ) i^i ( I ` ( f ` k ) ) ) ) ) )
49	42, 48	anbi12d 747	. . 3 |- ( ( G e. W /\ S e. NN0* ) -> ( ( f e. Word dom (iEdg ` G ) /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) ) <-> ( f e. Word dom I /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( I ` ( f ` ( k - 1 ) ) ) i^i ( I ` ( f ` k ) ) ) ) ) ) )
50	49	abbidv 2741	. 2 |- ( ( G e. W /\ S e. NN0* ) -> { f | ( f e. Word dom (iEdg ` G ) /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( (iEdg ` G ) ` ( f ` ( k - 1 ) ) ) i^i ( (iEdg ` G ) ` ( f ` k ) ) ) ) ) } = { f | ( f e. Word dom I /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( I ` ( f ` ( k - 1 ) ) ) i^i ( I ` ( f ` k ) ) ) ) ) } )
51	35, 50	eqtrd 2656	1 |- ( ( G e. W /\ S e. NN0* ) -> ( G EdgWalks S ) = { f | ( f e. Word dom I /\ A. k e. ( 1..^ ( # ` f ) ) S <_ ( # ` ( ( I ` ( f ` ( k - 1 ) ) ) i^i ( I ` ( f ` k ) ) ) ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1c1 9937   <_ cle 10075   - cmin 10266  NN0*cxnn0 11363  ..^cfzo 12465   #chash 13117  Word cword 13291  iEdgciedg 25875   EdgWalks cewlks 26491

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-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-ewlks 26494

This theorem is referenced by:  isewlk  26498