Theorem elwwlks2ons3 26848

Description: For each walk of length 2 between two vertices, there is a third vertex in the middle of the walk. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.)

Hypothesis

Ref	Expression
elwwlks2ons3.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
elwwlks2ons3	|- ( ( G e. U /\ A e. V /\ C e. V ) -> ( W e. ( A ( 2 WWalksNOn G ) C ) <-> E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) )

Distinct variable groups:   A, b   C, b   G, b   U, b   V, b   W, b

Proof of Theorem elwwlks2ons3

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 |- ( ( ( G e. U /\ A e. V /\ C e. V ) /\ W e. ( A ( 2 WWalksNOn G ) C ) ) -> W e. ( A ( 2 WWalksNOn G ) C ) )
2		elwwlks2ons3.v	. . . . . . . . 9 |- V = (Vtx ` G )
3	2	wwlknon 26742	. . . . . . . 8 |- ( ( A e. V /\ C e. V ) -> ( W e. ( A ( 2 WWalksNOn G ) C ) <-> ( W e. ( 2 WWalksN G ) /\ ( W ` 0 ) = A /\ ( W ` 2 ) = C ) ) )
4	3	3adant1 1079	. . . . . . 7 |- ( ( G e. U /\ A e. V /\ C e. V ) -> ( W e. ( A ( 2 WWalksNOn G ) C ) <-> ( W e. ( 2 WWalksN G ) /\ ( W ` 0 ) = A /\ ( W ` 2 ) = C ) ) )
5		wwlknbp2 26752	. . . . . . . . . 10 |- ( W e. ( 2 WWalksN G ) -> ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( 2 + 1 ) ) )
6		2p1e3 11151	. . . . . . . . . . . 12 |- ( 2 + 1 ) = 3
7	6	eqeq2i 2634	. . . . . . . . . . 11 |- ( ( # ` W ) = ( 2 + 1 ) <-> ( # ` W ) = 3 )
8		1ex 10035	. . . . . . . . . . . . . . . . 17 |- 1 e. _V
9	8	tpid2 4304	. . . . . . . . . . . . . . . 16 |- 1 e. { 0 , 1 , 2 }
10		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( ( # ` W ) = 3 -> ( 0..^ ( # ` W ) ) = ( 0..^ 3 ) )
11		fzo0to3tp 12554	. . . . . . . . . . . . . . . . 17 |- ( 0..^ 3 ) = { 0 , 1 , 2 }
12	10, 11	syl6eq 2672	. . . . . . . . . . . . . . . 16 |- ( ( # ` W ) = 3 -> ( 0..^ ( # ` W ) ) = { 0 , 1 , 2 } )
13	9, 12	syl5eleqr 2708	. . . . . . . . . . . . . . 15 |- ( ( # ` W ) = 3 -> 1 e. ( 0..^ ( # ` W ) ) )
14		wrdsymbcl 13318	. . . . . . . . . . . . . . 15 |- ( ( W e. Word (Vtx ` G ) /\ 1 e. ( 0..^ ( # ` W ) ) ) -> ( W ` 1 ) e. (Vtx ` G ) )
15	13, 14	sylan2 491	. . . . . . . . . . . . . 14 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) -> ( W ` 1 ) e. (Vtx ` G ) )
16	15	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) -> ( W ` 1 ) e. (Vtx ` G ) )
17		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) -> ( # ` W ) = 3 )
18	17	3ad2ant1 1082	. . . . . . . . . . . . . . . 16 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) -> ( # ` W ) = 3 )
19	18	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) /\ ( W ` 1 ) e. (Vtx ` G ) ) -> ( # ` W ) = 3 )
20		simpl 473	. . . . . . . . . . . . . . . . . 18 |- ( ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) -> ( W ` 0 ) = A )
21		eqidd 2623	. . . . . . . . . . . . . . . . . 18 |- ( ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) -> ( W ` 1 ) = ( W ` 1 ) )
22		simpr 477	. . . . . . . . . . . . . . . . . 18 |- ( ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) -> ( W ` 2 ) = C )
23	20, 21, 22	3jca 1242	. . . . . . . . . . . . . . . . 17 |- ( ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) -> ( ( W ` 0 ) = A /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = C ) )
24	23	3ad2ant2 1083	. . . . . . . . . . . . . . . 16 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) -> ( ( W ` 0 ) = A /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = C ) )
25	24	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) /\ ( W ` 1 ) e. (Vtx ` G ) ) -> ( ( W ` 0 ) = A /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = C ) )
26	2	eqcomi 2631	. . . . . . . . . . . . . . . . . . . . . 22 |- (Vtx ` G ) = V
27	26	wrdeqi 13328	. . . . . . . . . . . . . . . . . . . . 21 |- Word (Vtx ` G ) = Word V
28	27	eleq2i 2693	. . . . . . . . . . . . . . . . . . . 20 |- ( W e. Word (Vtx ` G ) <-> W e. Word V )
29	28	biimpi 206	. . . . . . . . . . . . . . . . . . 19 |- ( W e. Word (Vtx ` G ) -> W e. Word V )
30	29	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) -> W e. Word V )
31	30	3ad2ant1 1082	. . . . . . . . . . . . . . . . 17 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) -> W e. Word V )
32	31	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) /\ ( W ` 1 ) e. (Vtx ` G ) ) -> W e. Word V )
33		simpl32 1143	. . . . . . . . . . . . . . . 16 |- ( ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) /\ ( W ` 1 ) e. (Vtx ` G ) ) -> A e. V )
34	26	eleq2i 2693	. . . . . . . . . . . . . . . . . 18 |- ( ( W ` 1 ) e. (Vtx ` G ) <-> ( W ` 1 ) e. V )
35	34	biimpi 206	. . . . . . . . . . . . . . . . 17 |- ( ( W ` 1 ) e. (Vtx ` G ) -> ( W ` 1 ) e. V )
36	35	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) /\ ( W ` 1 ) e. (Vtx ` G ) ) -> ( W ` 1 ) e. V )
37		simpl33 1144	. . . . . . . . . . . . . . . 16 |- ( ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) /\ ( W ` 1 ) e. (Vtx ` G ) ) -> C e. V )
38		eqwrds3 13704	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word V /\ ( A e. V /\ ( W ` 1 ) e. V /\ C e. V ) ) -> ( W = <" A ( W ` 1 ) C "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = A /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = C ) ) ) )
39	32, 33, 36, 37, 38	syl13anc 1328	. . . . . . . . . . . . . . 15 |- ( ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) /\ ( W ` 1 ) e. (Vtx ` G ) ) -> ( W = <" A ( W ` 1 ) C "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = A /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = C ) ) ) )
40	19, 25, 39	mpbir2and 957	. . . . . . . . . . . . . 14 |- ( ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) /\ ( W ` 1 ) e. (Vtx ` G ) ) -> W = <" A ( W ` 1 ) C "> )
41	40, 36	jca 554	. . . . . . . . . . . . 13 |- ( ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) /\ ( W ` 1 ) e. (Vtx ` G ) ) -> ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) )
42	16, 41	mpdan 702	. . . . . . . . . . . 12 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) /\ ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) /\ ( G e. U /\ A e. V /\ C e. V ) ) -> ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) )
43	42	3exp 1264	. . . . . . . . . . 11 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = 3 ) -> ( ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) -> ( ( G e. U /\ A e. V /\ C e. V ) -> ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) ) ) )
44	7, 43	sylan2b 492	. . . . . . . . . 10 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( 2 + 1 ) ) -> ( ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) -> ( ( G e. U /\ A e. V /\ C e. V ) -> ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) ) ) )
45	5, 44	syl 17	. . . . . . . . 9 |- ( W e. ( 2 WWalksN G ) -> ( ( ( W ` 0 ) = A /\ ( W ` 2 ) = C ) -> ( ( G e. U /\ A e. V /\ C e. V ) -> ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) ) ) )
46	45	3impib 1262	. . . . . . . 8 |- ( ( W e. ( 2 WWalksN G ) /\ ( W ` 0 ) = A /\ ( W ` 2 ) = C ) -> ( ( G e. U /\ A e. V /\ C e. V ) -> ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) ) )
47	46	com12 32	. . . . . . 7 |- ( ( G e. U /\ A e. V /\ C e. V ) -> ( ( W e. ( 2 WWalksN G ) /\ ( W ` 0 ) = A /\ ( W ` 2 ) = C ) -> ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) ) )
48	4, 47	sylbid 230	. . . . . 6 |- ( ( G e. U /\ A e. V /\ C e. V ) -> ( W e. ( A ( 2 WWalksNOn G ) C ) -> ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) ) )
49	48	imp 445	. . . . 5 |- ( ( ( G e. U /\ A e. V /\ C e. V ) /\ W e. ( A ( 2 WWalksNOn G ) C ) ) -> ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) )
50		anass 681	. . . . 5 |- ( ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) <-> ( W e. ( A ( 2 WWalksNOn G ) C ) /\ ( W = <" A ( W ` 1 ) C "> /\ ( W ` 1 ) e. V ) ) )
51	1, 49, 50	sylanbrc 698	. . . 4 |- ( ( ( G e. U /\ A e. V /\ C e. V ) /\ W e. ( A ( 2 WWalksNOn G ) C ) ) -> ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) )
52		simpr 477	. . . . 5 |- ( ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) -> ( W ` 1 ) e. V )
53		eqidd 2623	. . . . . . . 8 |- ( b = ( W ` 1 ) -> A = A )
54		id 22	. . . . . . . 8 |- ( b = ( W ` 1 ) -> b = ( W ` 1 ) )
55		eqidd 2623	. . . . . . . 8 |- ( b = ( W ` 1 ) -> C = C )
56	53, 54, 55	s3eqd 13609	. . . . . . 7 |- ( b = ( W ` 1 ) -> <" A b C "> = <" A ( W ` 1 ) C "> )
57		eqeq2 2633	. . . . . . . 8 |- ( <" A b C "> = <" A ( W ` 1 ) C "> -> ( W = <" A b C "> <-> W = <" A ( W ` 1 ) C "> ) )
58		eleq1 2689	. . . . . . . 8 |- ( <" A b C "> = <" A ( W ` 1 ) C "> -> ( <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) <-> <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) )
59	57, 58	anbi12d 747	. . . . . . 7 |- ( <" A b C "> = <" A ( W ` 1 ) C "> -> ( ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) <-> ( W = <" A ( W ` 1 ) C "> /\ <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) )
60	56, 59	syl 17	. . . . . 6 |- ( b = ( W ` 1 ) -> ( ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) <-> ( W = <" A ( W ` 1 ) C "> /\ <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) )
61	60	adantl 482	. . . . 5 |- ( ( ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) /\ b = ( W ` 1 ) ) -> ( ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) <-> ( W = <" A ( W ` 1 ) C "> /\ <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) )
62		simpr 477	. . . . . . 7 |- ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) -> W = <" A ( W ` 1 ) C "> )
63		eleq1 2689	. . . . . . . 8 |- ( W = <" A ( W ` 1 ) C "> -> ( W e. ( A ( 2 WWalksNOn G ) C ) <-> <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) )
64	63	biimpac 503	. . . . . . 7 |- ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) -> <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) )
65	62, 64	jca 554	. . . . . 6 |- ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) -> ( W = <" A ( W ` 1 ) C "> /\ <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) )
66	65	adantr 481	. . . . 5 |- ( ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) -> ( W = <" A ( W ` 1 ) C "> /\ <" A ( W ` 1 ) C "> e. ( A ( 2 WWalksNOn G ) C ) ) )
67	52, 61, 66	rspcedvd 3317	. . . 4 |- ( ( ( W e. ( A ( 2 WWalksNOn G ) C ) /\ W = <" A ( W ` 1 ) C "> ) /\ ( W ` 1 ) e. V ) -> E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) )
68	51, 67	syl 17	. . 3 |- ( ( ( G e. U /\ A e. V /\ C e. V ) /\ W e. ( A ( 2 WWalksNOn G ) C ) ) -> E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) )
69	68	ex 450	. 2 |- ( ( G e. U /\ A e. V /\ C e. V ) -> ( W e. ( A ( 2 WWalksNOn G ) C ) -> E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) )
70		eleq1 2689	. . . . . 6 |- ( <" A b C "> = W -> ( <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) <-> W e. ( A ( 2 WWalksNOn G ) C ) ) )
71	70	eqcoms 2630	. . . . 5 |- ( W = <" A b C "> -> ( <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) <-> W e. ( A ( 2 WWalksNOn G ) C ) ) )
72	71	biimpa 501	. . . 4 |- ( ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) -> W e. ( A ( 2 WWalksNOn G ) C ) )
73	72	a1i 11	. . 3 |- ( ( G e. U /\ A e. V /\ C e. V ) -> ( ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) -> W e. ( A ( 2 WWalksNOn G ) C ) ) )
74	73	rexlimdvw 3034	. 2 |- ( ( G e. U /\ A e. V /\ C e. V ) -> ( E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) -> W e. ( A ( 2 WWalksNOn G ) C ) ) )
75	69, 74	impbid 202	1 |- ( ( G e. U /\ A e. V /\ C e. V ) -> ( W e. ( A ( 2 WWalksNOn G ) C ) <-> E. b e. V ( W = <" A b C "> /\ <" A b C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {ctp 4181   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   3c3 11071  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs3 13587  Vtxcvtx 25874   WWalksN cwwlksn 26718   WWalksNOn cwwlksnon 26719

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-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-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724

This theorem is referenced by:  elwwlks2on  26852  frgr2wwlk1  27193