Theorem clwwlksvbij 26922

Description: There is a bijection between the set of closed walks of a fixed length starting at a fixed vertex represented by walks (as word) and the set of closed walks (as words) of a fixed length starting at a fixed vertex. The difference between these two representations is that in the first case the starting vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.)

Assertion

Ref	Expression
clwwlksvbij	|- ( N e. NN -> E. f f : { w e. ( N WWalksN G ) | ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } )

Distinct variable groups:   w, G   w, N   f, G, w   f, N   S, f, w

Proof of Theorem clwwlksvbij

Dummy variables i x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . . . . 6 |- ( N WWalksN G ) e. _V
2	1	mptrabex 6488	. . . . 5 |- ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } |-> ( w substr <. 0 , N >. ) ) e. _V
3	2	resex 5443	. . . 4 |- ( ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } |-> ( w substr <. 0 , N >. ) ) |` { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } ) e. _V
4		eqid 2622	. . . . 5 |- ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } |-> ( w substr <. 0 , N >. ) ) = ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } |-> ( w substr <. 0 , N >. ) )
5		eqid 2622	. . . . . 6 |- { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } = { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) }
6	5, 4	clwwlksf1o 26919	. . . . 5 |- ( N e. NN -> ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } |-> ( w substr <. 0 , N >. ) ) : { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } -1-1-onto-> ( N ClWWalksN G ) )
7		fveq1 6190	. . . . . . . 8 |- ( y = ( w substr <. 0 , N >. ) -> ( y ` 0 ) = ( ( w substr <. 0 , N >. ) ` 0 ) )
8	7	eqeq1d 2624	. . . . . . 7 |- ( y = ( w substr <. 0 , N >. ) -> ( ( y ` 0 ) = S <-> ( ( w substr <. 0 , N >. ) ` 0 ) = S ) )
9	8	3ad2ant3 1084	. . . . . 6 |- ( ( N e. NN /\ w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } /\ y = ( w substr <. 0 , N >. ) ) -> ( ( y ` 0 ) = S <-> ( ( w substr <. 0 , N >. ) ` 0 ) = S ) )
10		fveq2 6191	. . . . . . . . . . . . 13 |- ( x = w -> ( lastS ` x ) = ( lastS ` w ) )
11		fveq1 6190	. . . . . . . . . . . . 13 |- ( x = w -> ( x ` 0 ) = ( w ` 0 ) )
12	10, 11	eqeq12d 2637	. . . . . . . . . . . 12 |- ( x = w -> ( ( lastS ` x ) = ( x ` 0 ) <-> ( lastS ` w ) = ( w ` 0 ) ) )
13	12	elrab 3363	. . . . . . . . . . 11 |- ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } <-> ( w e. ( N WWalksN G ) /\ ( lastS ` w ) = ( w ` 0 ) ) )
14		eqid 2622	. . . . . . . . . . . . . 14 |- (Vtx ` G ) = (Vtx ` G )
15		eqid 2622	. . . . . . . . . . . . . 14 |- (Edg ` G ) = (Edg ` G )
16	14, 15	wwlknp 26734	. . . . . . . . . . . . 13 |- ( w e. ( N WWalksN G ) -> ( w e. Word (Vtx ` G ) /\ ( # ` w ) = ( N + 1 ) /\ A. i e. ( 0..^ N ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) )
17		simpll 790	. . . . . . . . . . . . . . . 16 |- ( ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = ( N + 1 ) ) /\ N e. NN ) -> w e. Word (Vtx ` G ) )
18		nnz 11399	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN -> N e. ZZ )
19		uzid 11702	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
20		peano2uz 11741	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ( ZZ>= ` N ) -> ( N + 1 ) e. ( ZZ>= ` N ) )
21	18, 19, 20	3syl 18	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN -> ( N + 1 ) e. ( ZZ>= ` N ) )
22		elfz1end 12371	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN <-> N e. ( 1 ... N ) )
23	22	biimpi 206	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN -> N e. ( 1 ... N ) )
24		fzss2 12381	. . . . . . . . . . . . . . . . . . . 20 |- ( ( N + 1 ) e. ( ZZ>= ` N ) -> ( 1 ... N ) C_ ( 1 ... ( N + 1 ) ) )
25	24	sselda 3603	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N + 1 ) e. ( ZZ>= ` N ) /\ N e. ( 1 ... N ) ) -> N e. ( 1 ... ( N + 1 ) ) )
26	21, 23, 25	syl2anc 693	. . . . . . . . . . . . . . . . . 18 |- ( N e. NN -> N e. ( 1 ... ( N + 1 ) ) )
27	26	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = ( N + 1 ) ) /\ N e. NN ) -> N e. ( 1 ... ( N + 1 ) ) )
28		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` w ) = ( N + 1 ) -> ( 1 ... ( # ` w ) ) = ( 1 ... ( N + 1 ) ) )
29	28	eleq2d 2687	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` w ) = ( N + 1 ) -> ( N e. ( 1 ... ( # ` w ) ) <-> N e. ( 1 ... ( N + 1 ) ) ) )
30	29	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = ( N + 1 ) ) -> ( N e. ( 1 ... ( # ` w ) ) <-> N e. ( 1 ... ( N + 1 ) ) ) )
31	30	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = ( N + 1 ) ) /\ N e. NN ) -> ( N e. ( 1 ... ( # ` w ) ) <-> N e. ( 1 ... ( N + 1 ) ) ) )
32	27, 31	mpbird 247	. . . . . . . . . . . . . . . 16 |- ( ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = ( N + 1 ) ) /\ N e. NN ) -> N e. ( 1 ... ( # ` w ) ) )
33	17, 32	jca 554	. . . . . . . . . . . . . . 15 |- ( ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = ( N + 1 ) ) /\ N e. NN ) -> ( w e. Word (Vtx ` G ) /\ N e. ( 1 ... ( # ` w ) ) ) )
34	33	ex 450	. . . . . . . . . . . . . 14 |- ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = ( N + 1 ) ) -> ( N e. NN -> ( w e. Word (Vtx ` G ) /\ N e. ( 1 ... ( # ` w ) ) ) ) )
35	34	3adant3 1081	. . . . . . . . . . . . 13 |- ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = ( N + 1 ) /\ A. i e. ( 0..^ N ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( N e. NN -> ( w e. Word (Vtx ` G ) /\ N e. ( 1 ... ( # ` w ) ) ) ) )
36	16, 35	syl 17	. . . . . . . . . . . 12 |- ( w e. ( N WWalksN G ) -> ( N e. NN -> ( w e. Word (Vtx ` G ) /\ N e. ( 1 ... ( # ` w ) ) ) ) )
37	36	adantr 481	. . . . . . . . . . 11 |- ( ( w e. ( N WWalksN G ) /\ ( lastS ` w ) = ( w ` 0 ) ) -> ( N e. NN -> ( w e. Word (Vtx ` G ) /\ N e. ( 1 ... ( # ` w ) ) ) ) )
38	13, 37	sylbi 207	. . . . . . . . . 10 |- ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } -> ( N e. NN -> ( w e. Word (Vtx ` G ) /\ N e. ( 1 ... ( # ` w ) ) ) ) )
39	38	impcom 446	. . . . . . . . 9 |- ( ( N e. NN /\ w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } ) -> ( w e. Word (Vtx ` G ) /\ N e. ( 1 ... ( # ` w ) ) ) )
40		swrd0fv0 13440	. . . . . . . . 9 |- ( ( w e. Word (Vtx ` G ) /\ N e. ( 1 ... ( # ` w ) ) ) -> ( ( w substr <. 0 , N >. ) ` 0 ) = ( w ` 0 ) )
41	39, 40	syl 17	. . . . . . . 8 |- ( ( N e. NN /\ w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } ) -> ( ( w substr <. 0 , N >. ) ` 0 ) = ( w ` 0 ) )
42	41	eqeq1d 2624	. . . . . . 7 |- ( ( N e. NN /\ w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } ) -> ( ( ( w substr <. 0 , N >. ) ` 0 ) = S <-> ( w ` 0 ) = S ) )
43	42	3adant3 1081	. . . . . 6 |- ( ( N e. NN /\ w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } /\ y = ( w substr <. 0 , N >. ) ) -> ( ( ( w substr <. 0 , N >. ) ` 0 ) = S <-> ( w ` 0 ) = S ) )
44	9, 43	bitrd 268	. . . . 5 |- ( ( N e. NN /\ w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } /\ y = ( w substr <. 0 , N >. ) ) -> ( ( y ` 0 ) = S <-> ( w ` 0 ) = S ) )
45	4, 6, 44	f1oresrab 6395	. . . 4 |- ( N e. NN -> ( ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } |-> ( w substr <. 0 , N >. ) ) |` { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } ) : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { y e. ( N ClWWalksN G ) | ( y ` 0 ) = S } )
46		f1oeq1 6127	. . . . 5 |- ( f = ( ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } |-> ( w substr <. 0 , N >. ) ) |` { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } ) -> ( f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { y e. ( N ClWWalksN G ) | ( y ` 0 ) = S } <-> ( ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } |-> ( w substr <. 0 , N >. ) ) |` { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } ) : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { y e. ( N ClWWalksN G ) | ( y ` 0 ) = S } ) )
47	46	spcegv 3294	. . . 4 |- ( ( ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } |-> ( w substr <. 0 , N >. ) ) |` { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } ) e. _V -> ( ( ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } |-> ( w substr <. 0 , N >. ) ) |` { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } ) : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { y e. ( N ClWWalksN G ) | ( y ` 0 ) = S } -> E. f f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { y e. ( N ClWWalksN G ) | ( y ` 0 ) = S } ) )
48	3, 45, 47	mpsyl 68	. . 3 |- ( N e. NN -> E. f f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { y e. ( N ClWWalksN G ) | ( y ` 0 ) = S } )
49		fveq1 6190	. . . . . . 7 |- ( w = y -> ( w ` 0 ) = ( y ` 0 ) )
50	49	eqeq1d 2624	. . . . . 6 |- ( w = y -> ( ( w ` 0 ) = S <-> ( y ` 0 ) = S ) )
51	50	cbvrabv 3199	. . . . 5 |- { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } = { y e. ( N ClWWalksN G ) | ( y ` 0 ) = S }
52		f1oeq3 6129	. . . . 5 |- ( { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } = { y e. ( N ClWWalksN G ) | ( y ` 0 ) = S } -> ( f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } <-> f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { y e. ( N ClWWalksN G ) | ( y ` 0 ) = S } ) )
53	51, 52	mp1i 13	. . . 4 |- ( N e. NN -> ( f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } <-> f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { y e. ( N ClWWalksN G ) | ( y ` 0 ) = S } ) )
54	53	exbidv 1850	. . 3 |- ( N e. NN -> ( E. f f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } <-> E. f f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { y e. ( N ClWWalksN G ) | ( y ` 0 ) = S } ) )
55	48, 54	mpbird 247	. 2 |- ( N e. NN -> E. f f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } )
56		df-rab 2921	. . . . 5 |- { w e. ( N WWalksN G ) | ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) } = { w | ( w e. ( N WWalksN G ) /\ ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) ) }
57		anass 681	. . . . . . 7 |- ( ( ( w e. ( N WWalksN G ) /\ ( lastS ` w ) = ( w ` 0 ) ) /\ ( w ` 0 ) = S ) <-> ( w e. ( N WWalksN G ) /\ ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) ) )
58	57	bicomi 214	. . . . . 6 |- ( ( w e. ( N WWalksN G ) /\ ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) ) <-> ( ( w e. ( N WWalksN G ) /\ ( lastS ` w ) = ( w ` 0 ) ) /\ ( w ` 0 ) = S ) )
59	58	abbii 2739	. . . . 5 |- { w | ( w e. ( N WWalksN G ) /\ ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) ) } = { w | ( ( w e. ( N WWalksN G ) /\ ( lastS ` w ) = ( w ` 0 ) ) /\ ( w ` 0 ) = S ) }
60	13	bicomi 214	. . . . . . . 8 |- ( ( w e. ( N WWalksN G ) /\ ( lastS ` w ) = ( w ` 0 ) ) <-> w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } )
61	60	anbi1i 731	. . . . . . 7 |- ( ( ( w e. ( N WWalksN G ) /\ ( lastS ` w ) = ( w ` 0 ) ) /\ ( w ` 0 ) = S ) <-> ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } /\ ( w ` 0 ) = S ) )
62	61	abbii 2739	. . . . . 6 |- { w | ( ( w e. ( N WWalksN G ) /\ ( lastS ` w ) = ( w ` 0 ) ) /\ ( w ` 0 ) = S ) } = { w | ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } /\ ( w ` 0 ) = S ) }
63		df-rab 2921	. . . . . 6 |- { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } = { w | ( w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } /\ ( w ` 0 ) = S ) }
64	62, 63	eqtr4i 2647	. . . . 5 |- { w | ( ( w e. ( N WWalksN G ) /\ ( lastS ` w ) = ( w ` 0 ) ) /\ ( w ` 0 ) = S ) } = { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S }
65	56, 59, 64	3eqtri 2648	. . . 4 |- { w e. ( N WWalksN G ) | ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) } = { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S }
66		f1oeq2 6128	. . . 4 |- ( { w e. ( N WWalksN G ) | ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) } = { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -> ( f : { w e. ( N WWalksN G ) | ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } <-> f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } ) )
67	65, 66	mp1i 13	. . 3 |- ( N e. NN -> ( f : { w e. ( N WWalksN G ) | ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } <-> f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } ) )
68	67	exbidv 1850	. 2 |- ( N e. NN -> ( E. f f : { w e. ( N WWalksN G ) | ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } <-> E. f f : { w e. { x e. ( N WWalksN G ) | ( lastS ` x ) = ( x ` 0 ) } | ( w ` 0 ) = S } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } ) )
69	55, 68	mpbird 247	1 |- ( N e. NN -> E. f f : { w e. ( N WWalksN G ) | ( ( lastS ` w ) = ( w ` 0 ) /\ ( w ` 0 ) = S ) } -1-1-onto-> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = S } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   _Vcvv 3200   {cpr 4179   <.cop 4183   |-> cmpt 4729   |` cres 5116   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939   WWalksN cwwlksn 26718   ClWWalksN cclwwlksn 26876

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-fal 1489  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-oadd 7564  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-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-wwlks 26722  df-wwlksn 26723  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwwlkqhash  27233