Theorem el2wlkonot 30393

Description: A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)

Assertion

Ref	Expression
el2wlkonot	|- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) <-> E. b e. V ( T = <. A , b , C >. /\ E. f E. p ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )

Distinct variable groups:   A, b, f, p   C, b, f, p   E, b, f, p   T, b, f, p   V, b, f, p   X, b, f, p   Y, b, f, p

Proof of Theorem el2wlkonot

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2wlkonot 30389	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( A ( V 2WalksOnOt E ) C ) = { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = A /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = C ) ) } )
2	1	eleq2d 2510	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) <-> T e. { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = A /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = C ) ) } ) )
3		fveq2 5696	. . . . . . . . 9 |- ( t = T -> ( 1st ` t ) = ( 1st ` T ) )
4	3	fveq2d 5700	. . . . . . . 8 |- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) )
5	4	eqeq1d 2451	. . . . . . 7 |- ( t = T -> ( ( 1st ` ( 1st ` t ) ) = A <-> ( 1st ` ( 1st ` T ) ) = A ) )
6	3	fveq2d 5700	. . . . . . . 8 |- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) )
7	6	eqeq1d 2451	. . . . . . 7 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) <-> ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) ) )
8		fveq2 5696	. . . . . . . 8 |- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) )
9	8	eqeq1d 2451	. . . . . . 7 |- ( t = T -> ( ( 2nd ` t ) = C <-> ( 2nd ` T ) = C ) )
10	5, 7, 9	3anbi123d 1289	. . . . . 6 |- ( t = T -> ( ( ( 1st ` ( 1st ` t ) ) = A /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = C ) <-> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) )
11	10	3anbi3d 1295	. . . . 5 |- ( t = T -> ( ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = A /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = C ) ) <-> ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) )
12	11	2exbidv 1682	. . . 4 |- ( t = T -> ( E. f E. p ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = A /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = C ) ) <-> E. f E. p ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) )
13	12	elrab 3122	. . 3 |- ( T e. { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = A /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = C ) ) } <-> ( T e. ( ( V X. V ) X. V ) /\ E. f E. p ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) )
14	2, 13	syl6bb 261	. 2 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) <-> ( T e. ( ( V X. V ) X. V ) /\ E. f E. p ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) ) )
15		simpl 457	. . . . . . . . 9 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( V e. X /\ E e. Y ) )
16		vex 2980	. . . . . . . . . . 11 |- f e. _V
17		vex 2980	. . . . . . . . . . 11 |- p e. _V
18	16, 17	pm3.2i 455	. . . . . . . . . 10 |- ( f e. _V /\ p e. _V )
19	18	a1i 11	. . . . . . . . 9 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( f e. _V /\ p e. _V ) )
20		simpr 461	. . . . . . . . 9 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( A e. V /\ C e. V ) )
21		iswlkon 23435	. . . . . . . . 9 |- ( ( ( V e. X /\ E e. Y ) /\ ( f e. _V /\ p e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( f ( A ( V WalkOn E ) C ) p <-> ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) ) )
22	15, 19, 20, 21	syl3anc 1218	. . . . . . . 8 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( f ( A ( V WalkOn E ) C ) p <-> ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) ) )
23	22	3anbi1d 1293	. . . . . . 7 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) <-> ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) )
24	23	anbi2d 703	. . . . . 6 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( ( T e. ( ( V X. V ) X. V ) /\ ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) <-> ( T e. ( ( V X. V ) X. V ) /\ ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) ) )
25		simpl 457	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A e. V /\ C e. V ) -> A e. V )
26	25	adantl 466	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> A e. V )
27	26	adantl 466	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) -> A e. V )
28	27	adantr 465	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) /\ ( # ` f ) = 2 ) -> A e. V )
29		el2wlkonotlem 30386	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> ( p ` 1 ) e. V )
30	29	ex 434	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f ( V Walks E ) p -> ( ( # ` f ) = 2 -> ( p ` 1 ) e. V ) )
31	30	3ad2ant1 1009	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) -> ( ( # ` f ) = 2 -> ( p ` 1 ) e. V ) )
32	31	adantr 465	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) -> ( ( # ` f ) = 2 -> ( p ` 1 ) e. V ) )
33	32	imp 429	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) /\ ( # ` f ) = 2 ) -> ( p ` 1 ) e. V )
34		simpr 461	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A e. V /\ C e. V ) -> C e. V )
35	34	adantl 466	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> C e. V )
36	35	adantl 466	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) -> C e. V )
37	36	adantr 465	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) /\ ( # ` f ) = 2 ) -> C e. V )
38		el2xptp0 30132	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. V /\ ( p ` 1 ) e. V /\ C e. V ) -> ( ( T e. ( ( V X. V ) X. V ) /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) <-> T = <. A , ( p ` 1 ) , C >. ) )
39	28, 33, 37, 38	syl3anc 1218	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) /\ ( # ` f ) = 2 ) -> ( ( T e. ( ( V X. V ) X. V ) /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) <-> T = <. A , ( p ` 1 ) , C >. ) )
40		oteq2 4074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ( ( p ` 1 ) = b -> <. A , ( p ` 1 ) , C >. = <. A , b , C >. )
41	40	eqcoms 2446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ( b = ( p ` 1 ) -> <. A , ( p ` 1 ) , C >. = <. A , b , C >. )
42	41	eqeq2d 2454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( b = ( p ` 1 ) -> ( T = <. A , ( p ` 1 ) , C >. <-> T = <. A , b , C >. ) )
43	42	biimpcd 224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( T = <. A , ( p ` 1 ) , C >. -> ( b = ( p ` 1 ) -> T = <. A , b , C >. ) )
44	43	adantr 465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) -> ( b = ( p ` 1 ) -> T = <. A , b , C >. ) )
45	44	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) -> ( b = ( p ` 1 ) -> T = <. A , b , C >. ) )
46	45	impcom 430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( b = ( p ` 1 ) /\ ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) ) -> T = <. A , b , C >. )
47		simpl 457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> f ( V Walks E ) p )
48	47	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) -> f ( V Walks E ) p )
49	48	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) -> f ( V Walks E ) p )
50	49	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( b = ( p ` 1 ) /\ ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) ) -> f ( V Walks E ) p )
51		simpr 461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> ( # ` f ) = 2 )
52	51	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) -> ( # ` f ) = 2 )
53	52	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) -> ( # ` f ) = 2 )
54	53	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( b = ( p ` 1 ) /\ ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) ) -> ( # ` f ) = 2 )
55		eqcom 2445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ( ( p ` 0 ) = A <-> A = ( p ` 0 ) )
56	55	biimpi 194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( ( p ` 0 ) = A -> A = ( p ` 0 ) )
57	56	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) -> A = ( p ` 0 ) )
58	57	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( b = ( p ` 1 ) /\ ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) ) -> A = ( p ` 0 ) )
59		simpl 457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( b = ( p ` 1 ) /\ ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) ) -> b = ( p ` 1 ) )
60		fveq2 5696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ( ( # ` f ) = 2 -> ( p ` ( # ` f ) ) = ( p ` 2 ) )
61	60	eqeq1d 2451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- ( ( # ` f ) = 2 -> ( ( p ` ( # ` f ) ) = C <-> ( p ` 2 ) = C ) )
62		eqcom 2445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ( ( p ` 2 ) = C <-> C = ( p ` 2 ) )
63	62	biimpi 194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- ( ( p ` 2 ) = C -> C = ( p ` 2 ) )
64	61, 63	syl6bi 228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- ( ( # ` f ) = 2 -> ( ( p ` ( # ` f ) ) = C -> C = ( p ` 2 ) ) )
65	64	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> ( ( p ` ( # ` f ) ) = C -> C = ( p ` 2 ) ) )
66	65	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) -> ( ( p ` ( # ` f ) ) = C -> C = ( p ` 2 ) ) )
67	66	imp 429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) -> C = ( p ` 2 ) )
68	67	adantr 465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) -> C = ( p ` 2 ) )
69	68	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( b = ( p ` 1 ) /\ ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) ) -> C = ( p ` 2 ) )
70	58, 59, 69	3jca 1168	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( b = ( p ` 1 ) /\ ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) ) -> ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) )
71	50, 54, 70	3jca 1168	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( b = ( p ` 1 ) /\ ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) ) -> ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) )
72	46, 71	jca 532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( b = ( p ` 1 ) /\ ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) ) -> ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) )
73	72	ex 434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( b = ( p ` 1 ) -> ( ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) -> ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
74	73	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) /\ b = ( p ` 1 ) ) -> ( ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) -> ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
75	29, 74	rspcimedv 3080	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> ( ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
76	75	com12 31	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) -> ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
77	76	exp41 610	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( T = <. A , ( p ` 1 ) , C >. -> ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> ( ( p ` ( # ` f ) ) = C -> ( ( p ` 0 ) = A -> ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) ) ) )
78	77	com15 93	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> ( ( p ` ( # ` f ) ) = C -> ( ( p ` 0 ) = A -> ( T = <. A , ( p ` 1 ) , C >. -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) ) ) )
79	78	pm2.43i 47	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> ( ( p ` ( # ` f ) ) = C -> ( ( p ` 0 ) = A -> ( T = <. A , ( p ` 1 ) , C >. -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) ) )
80	79	ex 434	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f ( V Walks E ) p -> ( ( # ` f ) = 2 -> ( ( p ` ( # ` f ) ) = C -> ( ( p ` 0 ) = A -> ( T = <. A , ( p ` 1 ) , C >. -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) ) ) )
81	80	com24 87	. . . . . . . . . . . . . . . . . . . . 21 |- ( f ( V Walks E ) p -> ( ( p ` 0 ) = A -> ( ( p ` ( # ` f ) ) = C -> ( ( # ` f ) = 2 -> ( T = <. A , ( p ` 1 ) , C >. -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) ) ) )
82	81	3imp 1181	. . . . . . . . . . . . . . . . . . . 20 |- ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) -> ( ( # ` f ) = 2 -> ( T = <. A , ( p ` 1 ) , C >. -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) )
83	82	adantr 465	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) -> ( ( # ` f ) = 2 -> ( T = <. A , ( p ` 1 ) , C >. -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) )
84	83	imp 429	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) /\ ( # ` f ) = 2 ) -> ( T = <. A , ( p ` 1 ) , C >. -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
85	39, 84	sylbid 215	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) /\ ( # ` f ) = 2 ) -> ( ( T e. ( ( V X. V ) X. V ) /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
86	85	ex 434	. . . . . . . . . . . . . . . 16 |- ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) -> ( ( # ` f ) = 2 -> ( ( T e. ( ( V X. V ) X. V ) /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) )
87	86	com23 78	. . . . . . . . . . . . . . 15 |- ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) ) -> ( ( T e. ( ( V X. V ) X. V ) /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) -> ( ( # ` f ) = 2 -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) )
88	87	ex 434	. . . . . . . . . . . . . 14 |- ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) -> ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( ( T e. ( ( V X. V ) X. V ) /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) -> ( ( # ` f ) = 2 -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) ) )
89	88	com4t 85	. . . . . . . . . . . . 13 |- ( ( T e. ( ( V X. V ) X. V ) /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) -> ( ( # ` f ) = 2 -> ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) -> ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) ) )
90	89	ex 434	. . . . . . . . . . . 12 |- ( T e. ( ( V X. V ) X. V ) -> ( ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) -> ( ( # ` f ) = 2 -> ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) -> ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) ) ) )
91	90	com14 88	. . . . . . . . . . 11 |- ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) -> ( ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) -> ( ( # ` f ) = 2 -> ( T e. ( ( V X. V ) X. V ) -> ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) ) ) )
92	91	com23 78	. . . . . . . . . 10 |- ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) -> ( ( # ` f ) = 2 -> ( ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) -> ( T e. ( ( V X. V ) X. V ) -> ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) ) ) )
93	92	3imp 1181	. . . . . . . . 9 |- ( ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) -> ( T e. ( ( V X. V ) X. V ) -> ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) ) )
94	93	impcom 430	. . . . . . . 8 |- ( ( T e. ( ( V X. V ) X. V ) /\ ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) -> ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
95	94	com12 31	. . . . . . 7 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( ( T e. ( ( V X. V ) X. V ) /\ ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) -> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
96	25	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> A e. V )
97		simpr 461	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> b e. V )
98	34	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> C e. V )
99	96, 97, 98	3jca 1168	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> ( A e. V /\ b e. V /\ C e. V ) )
100		otel3xp 30133	. . . . . . . . . . . . . . . 16 |- ( ( T = <. A , b , C >. /\ ( A e. V /\ b e. V /\ C e. V ) ) -> T e. ( ( V X. V ) X. V ) )
101	99, 100	sylan2 474	. . . . . . . . . . . . . . 15 |- ( ( T = <. A , b , C >. /\ ( ( A e. V /\ C e. V ) /\ b e. V ) ) -> T e. ( ( V X. V ) X. V ) )
102	101	ex 434	. . . . . . . . . . . . . 14 |- ( T = <. A , b , C >. -> ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> T e. ( ( V X. V ) X. V ) ) )
103	102	adantr 465	. . . . . . . . . . . . 13 |- ( ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> T e. ( ( V X. V ) X. V ) ) )
104	103	com12 31	. . . . . . . . . . . 12 |- ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> ( ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> T e. ( ( V X. V ) X. V ) ) )
105	104	adantll 713	. . . . . . . . . . 11 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) -> ( ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> T e. ( ( V X. V ) X. V ) ) )
106	105	imp 429	. . . . . . . . . 10 |- ( ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) /\ ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) -> T e. ( ( V X. V ) X. V ) )
107		id 22	. . . . . . . . . . . . . . 15 |- ( f ( V Walks E ) p -> f ( V Walks E ) p )
108	107	3ad2ant1 1009	. . . . . . . . . . . . . 14 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> f ( V Walks E ) p )
109	108	adantl 466	. . . . . . . . . . . . 13 |- ( ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> f ( V Walks E ) p )
110	109	adantl 466	. . . . . . . . . . . 12 |- ( ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) /\ ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) -> f ( V Walks E ) p )
111		eqcom 2445	. . . . . . . . . . . . . . . . 17 |- ( A = ( p ` 0 ) <-> ( p ` 0 ) = A )
112	111	biimpi 194	. . . . . . . . . . . . . . . 16 |- ( A = ( p ` 0 ) -> ( p ` 0 ) = A )
113	112	3ad2ant1 1009	. . . . . . . . . . . . . . 15 |- ( ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p ` 0 ) = A )
114	113	3ad2ant3 1011	. . . . . . . . . . . . . 14 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( p ` 0 ) = A )
115	114	adantl 466	. . . . . . . . . . . . 13 |- ( ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( p ` 0 ) = A )
116	115	adantl 466	. . . . . . . . . . . 12 |- ( ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) /\ ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) -> ( p ` 0 ) = A )
117		eqcom 2445	. . . . . . . . . . . . . . . . . . 19 |- ( C = ( p ` 2 ) <-> ( p ` 2 ) = C )
118	117	biimpi 194	. . . . . . . . . . . . . . . . . 18 |- ( C = ( p ` 2 ) -> ( p ` 2 ) = C )
119	118	3ad2ant3 1011	. . . . . . . . . . . . . . . . 17 |- ( ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p ` 2 ) = C )
120	119, 61	syl5ibr 221	. . . . . . . . . . . . . . . 16 |- ( ( # ` f ) = 2 -> ( ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p ` ( # ` f ) ) = C ) )
121	120	a1i 11	. . . . . . . . . . . . . . 15 |- ( f ( V Walks E ) p -> ( ( # ` f ) = 2 -> ( ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p ` ( # ` f ) ) = C ) ) )
122	121	3imp 1181	. . . . . . . . . . . . . 14 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( p ` ( # ` f ) ) = C )
123	122	adantl 466	. . . . . . . . . . . . 13 |- ( ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( p ` ( # ` f ) ) = C )
124	123	adantl 466	. . . . . . . . . . . 12 |- ( ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) /\ ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) -> ( p ` ( # ` f ) ) = C )
125	110, 116, 124	3jca 1168	. . . . . . . . . . 11 |- ( ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) /\ ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) -> ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) )
126		id 22	. . . . . . . . . . . . . 14 |- ( ( # ` f ) = 2 -> ( # ` f ) = 2 )
127	126	3ad2ant2 1010	. . . . . . . . . . . . 13 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( # ` f ) = 2 )
128	127	adantl 466	. . . . . . . . . . . 12 |- ( ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( # ` f ) = 2 )
129	128	adantl 466	. . . . . . . . . . 11 |- ( ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) /\ ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) -> ( # ` f ) = 2 )
130	99	adantll 713	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) -> ( A e. V /\ b e. V /\ C e. V ) )
131		oteqimp 30134	. . . . . . . . . . . . . . . . 17 |- ( T = <. A , b , C >. -> ( ( A e. V /\ b e. V /\ C e. V ) -> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = b /\ ( 2nd ` T ) = C ) ) )
132	130, 131	syl5 32	. . . . . . . . . . . . . . . 16 |- ( T = <. A , b , C >. -> ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) -> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = b /\ ( 2nd ` T ) = C ) ) )
133		eqeq2 2452	. . . . . . . . . . . . . . . . . 18 |- ( b = ( p ` 1 ) -> ( ( 2nd ` ( 1st ` T ) ) = b <-> ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) ) )
134	133	3anbi2d 1294	. . . . . . . . . . . . . . . . 17 |- ( b = ( p ` 1 ) -> ( ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = b /\ ( 2nd ` T ) = C ) <-> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) )
135	134	imbi2d 316	. . . . . . . . . . . . . . . 16 |- ( b = ( p ` 1 ) -> ( ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) -> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = b /\ ( 2nd ` T ) = C ) ) <-> ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) -> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) )
136	132, 135	syl5ib 219	. . . . . . . . . . . . . . 15 |- ( b = ( p ` 1 ) -> ( T = <. A , b , C >. -> ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) -> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) )
137	136	3ad2ant2 1010	. . . . . . . . . . . . . 14 |- ( ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( T = <. A , b , C >. -> ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) -> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) )
138	137	3ad2ant3 1011	. . . . . . . . . . . . 13 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( T = <. A , b , C >. -> ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) -> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) )
139	138	impcom 430	. . . . . . . . . . . 12 |- ( ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) -> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) )
140	139	impcom 430	. . . . . . . . . . 11 |- ( ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) /\ ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) -> ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) )
141	125, 129, 140	3jca 1168	. . . . . . . . . 10 |- ( ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) /\ ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) -> ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) )
142	106, 141	jca 532	. . . . . . . . 9 |- ( ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) /\ ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) -> ( T e. ( ( V X. V ) X. V ) /\ ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) )
143	142	ex 434	. . . . . . . 8 |- ( ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) /\ b e. V ) -> ( ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( T e. ( ( V X. V ) X. V ) /\ ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) ) )
144	143	rexlimdva 2846	. . . . . . 7 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( T e. ( ( V X. V ) X. V ) /\ ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) ) )
145	95, 144	impbid 191	. . . . . 6 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( ( T e. ( ( V X. V ) X. V ) /\ ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) <-> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
146	24, 145	bitrd 253	. . . . 5 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( ( T e. ( ( V X. V ) X. V ) /\ ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) <-> E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
147	146	2exbidv 1682	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( E. f E. p ( T e. ( ( V X. V ) X. V ) /\ ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) <-> E. f E. p E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
148		19.42vv 1926	. . . 4 |- ( E. f E. p ( T e. ( ( V X. V ) X. V ) /\ ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) <-> ( T e. ( ( V X. V ) X. V ) /\ E. f E. p ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) )
149		rexcom4 2997	. . . . 5 |- ( E. b e. V E. f E. p ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) <-> E. f E. b e. V E. p ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) )
150		rexcom4 2997	. . . . . 6 |- ( E. b e. V E. p ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) <-> E. p E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) )
151	150	exbii 1634	. . . . 5 |- ( E. f E. b e. V E. p ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) <-> E. f E. p E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) )
152	149, 151	bitr2i 250	. . . 4 |- ( E. f E. p E. b e. V ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) <-> E. b e. V E. f E. p ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) )
153	147, 148, 152	3bitr3g 287	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( ( T e. ( ( V X. V ) X. V ) /\ E. f E. p ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) <-> E. b e. V E. f E. p ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
154		19.42vv 1926	. . . 4 |- ( E. f E. p ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) <-> ( T = <. A , b , C >. /\ E. f E. p ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) )
155	154	rexbii 2745	. . 3 |- ( E. b e. V E. f E. p ( T = <. A , b , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) <-> E. b e. V ( T = <. A , b , C >. /\ E. f E. p ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) )
156	153, 155	syl6bb 261	. 2 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( ( T e. ( ( V X. V ) X. V ) /\ E. f E. p ( f ( A ( V WalkOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` T ) ) = A /\ ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) /\ ( 2nd ` T ) = C ) ) ) <-> E. b e. V ( T = <. A , b , C >. /\ E. f E. p ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
157	14, 156	bitrd 253	1 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) <-> E. b e. V ( T = <. A , b , C >. /\ E. f E. p ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   E.wex 1586   e. wcel 1756   E.wrex 2721   {crab 2724   _Vcvv 2977   <.cotp 3890   class class class wbr 4297   X. cxp 4843   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   0cc0 9287   1c1 9288   2c2 10376   #chash 12108   Walks cwalk 23410   WalkOn cwlkon 23414   2WalksOnOt c2wlkonot 30379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-ot 3891  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-word 12234  df-wlk 23420  df-wlkon 23426  df-2wlkonot 30382

This theorem is referenced by:  el2wlkonotot0  30396  el2wlksot  30404  frg2wot1  30655  frg2woteq  30658