Theorem el2wlkonot 28066

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 28062	. . . 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 2471	. . 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 5687	. . . . . . . . 9 |- ( t = T -> ( 1st ` t ) = ( 1st ` T ) )
4	3	fveq2d 5691	. . . . . . . 8 |- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) )
5	4	eqeq1d 2412	. . . . . . 7 |- ( t = T -> ( ( 1st ` ( 1st ` t ) ) = A <-> ( 1st ` ( 1st ` T ) ) = A ) )
6	3	fveq2d 5691	. . . . . . . 8 |- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) )
7	6	eqeq1d 2412	. . . . . . 7 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) <-> ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) ) )
8		fveq2 5687	. . . . . . . 8 |- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) )
9	8	eqeq1d 2412	. . . . . . 7 |- ( t = T -> ( ( 2nd ` t ) = C <-> ( 2nd ` T ) = C ) )
10	5, 7, 9	3anbi123d 1254	. . . . . 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 1260	. . . . 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 1635	. . . 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 3052	. . 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 253	. 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 444	. . . . . . . . 9 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( V e. X /\ E e. Y ) )
16		vex 2919	. . . . . . . . . . 11 |- f e. _V
17		vex 2919	. . . . . . . . . . 11 |- p e. _V
18	16, 17	pm3.2i 442	. . . . . . . . . 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 448	. . . . . . . . 9 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> ( A e. V /\ C e. V ) )
21		iswlkon 21484	. . . . . . . . 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 1184	. . . . . . . 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 1258	. . . . . . 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 685	. . . . . 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 444	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A e. V /\ C e. V ) -> A e. V )
26	25	adantl 453	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> A e. V )
27	26	adantl 453	. . . . . . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . . . . . 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 28059	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> ( p ` 1 ) e. V )
30	29	ex 424	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f ( V Walks E ) p -> ( ( # ` f ) = 2 -> ( p ` 1 ) e. V ) )
31	30	3ad2ant1 978	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( f ( V Walks E ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) -> ( ( # ` f ) = 2 -> ( p ` 1 ) e. V ) )
32	31	adantr 452	. . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . 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 448	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A e. V /\ C e. V ) -> C e. V )
35	34	adantl 453	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ C e. V ) ) -> C e. V )
36	35	adantl 453	. . . . . . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . . . . . 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 27949	. . . . . . . . . . . . . . . . . . 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 1184	. . . . . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ( ( p ` 1 ) = b -> <. A , ( p ` 1 ) , C >. = <. A , b , C >. )
41	40	eqcoms 2407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ( b = ( p ` 1 ) -> <. A , ( p ` 1 ) , C >. = <. A , b , C >. )
42	41	eqeq2d 2415	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( b = ( p ` 1 ) -> ( T = <. A , ( p ` 1 ) , C >. <-> T = <. A , b , C >. ) )
43	42	biimpcd 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( T = <. A , ( p ` 1 ) , C >. -> ( b = ( p ` 1 ) -> T = <. A , b , C >. ) )
44	43	adantr 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) -> ( b = ( p ` 1 ) -> T = <. A , b , C >. ) )
45	44	ad2antrr 707	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> f ( V Walks E ) p )
48	47	adantl 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) -> f ( V Walks E ) p )
49	48	ad2antrr 707	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> ( # ` f ) = 2 )
52	51	adantl 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) -> ( # ` f ) = 2 )
53	52	ad2antrr 707	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) /\ ( p ` 0 ) = A ) -> ( # ` f ) = 2 )
54	53	adantl 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ( ( p ` 0 ) = A <-> A = ( p ` 0 ) )
56	55	biimpi 187	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( ( p ` 0 ) = A -> A = ( p ` 0 ) )
57	56	adantl 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ( ( # ` f ) = 2 -> ( p ` ( # ` f ) ) = ( p ` 2 ) )
61	60	eqeq1d 2412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- ( ( # ` f ) = 2 -> ( ( p ` ( # ` f ) ) = C <-> ( p ` 2 ) = C ) )
62		eqcom 2406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ( ( p ` 2 ) = C <-> C = ( p ` 2 ) )
63	62	biimpi 187	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- ( ( p ` 2 ) = C -> C = ( p ` 2 ) )
64	61, 63	syl6bi 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- ( ( # ` f ) = 2 -> ( ( p ` ( # ` f ) ) = C -> C = ( p ` 2 ) ) )
65	64	adantl 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) -> ( ( p ` ( # ` f ) ) = C -> C = ( p ` 2 ) ) )
66	65	adantl 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) -> ( ( p ` ( # ` f ) ) = C -> C = ( p ` 2 ) ) )
67	66	imp 419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ( ( ( T = <. A , ( p ` 1 ) , C >. /\ ( f ( V Walks E ) p /\ ( # ` f ) = 2 ) ) /\ ( p ` ( # ` f ) ) = C ) -> C = ( p ` 2 ) )
68	67	adantr 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 519	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 424	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3014	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 29	. . . . . . . . . . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . . . . . . . . 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 89	. . . . . . . . . . . . . . . . . . . . . . . 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 45	. . . . . . . . . . . . . . . . . . . . . . 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 424	. . . . . . . . . . . . . . . . . . . . . 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 83	. . . . . . . . . . . . . . . . . . . . 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 1147	. . . . . . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . 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 207	. . . . . . . . . . . . . . . . 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 424	. . . . . . . . . . . . . . . 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 74	. . . . . . . . . . . . . . 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 424	. . . . . . . . . . . . . 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 81	. . . . . . . . . . . . 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 424	. . . . . . . . . . . 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 84	. . . . . . . . . . 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 74	. . . . . . . . . 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 1147	. . . . . . . . 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 420	. . . . . . . 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 29	. . . . . . 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 452	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> A e. V )
97		simpr 448	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> b e. V )
98	34	adantr 452	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> C e. V )
99	96, 97, 98	3jca 1134	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> ( A e. V /\ b e. V /\ C e. V ) )
100		otel3xp 27950	. . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . 15 |- ( ( T = <. A , b , C >. /\ ( ( A e. V /\ C e. V ) /\ b e. V ) ) -> T e. ( ( V X. V ) X. V ) )
102	101	ex 424	. . . . . . . . . . . . . 14 |- ( T = <. A , b , C >. -> ( ( ( A e. V /\ C e. V ) /\ b e. V ) -> T e. ( ( V X. V ) X. V ) ) )
103	102	adantr 452	. . . . . . . . . . . . 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 29	. . . . . . . . . . . 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 695	. . . . . . . . . . 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 419	. . . . . . . . . 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 20	. . . . . . . . . . . . . . 15 |- ( f ( V Walks E ) p -> f ( V Walks E ) p )
108	107	3ad2ant1 978	. . . . . . . . . . . . . 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 453	. . . . . . . . . . . . 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 453	. . . . . . . . . . . 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 2406	. . . . . . . . . . . . . . . . 17 |- ( A = ( p ` 0 ) <-> ( p ` 0 ) = A )
112	111	biimpi 187	. . . . . . . . . . . . . . . 16 |- ( A = ( p ` 0 ) -> ( p ` 0 ) = A )
113	112	3ad2ant1 978	. . . . . . . . . . . . . . 15 |- ( ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p ` 0 ) = A )
114	113	3ad2ant3 980	. . . . . . . . . . . . . 14 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( p ` 0 ) = A )
115	114	adantl 453	. . . . . . . . . . . . 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 453	. . . . . . . . . . . 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 2406	. . . . . . . . . . . . . . . . . . 19 |- ( C = ( p ` 2 ) <-> ( p ` 2 ) = C )
118	117	biimpi 187	. . . . . . . . . . . . . . . . . 18 |- ( C = ( p ` 2 ) -> ( p ` 2 ) = C )
119	118	3ad2ant3 980	. . . . . . . . . . . . . . . . 17 |- ( ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p ` 2 ) = C )
120	119, 61	syl5ibr 213	. . . . . . . . . . . . . . . 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 1147	. . . . . . . . . . . . . 14 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( p ` ( # ` f ) ) = C )
123	122	adantl 453	. . . . . . . . . . . . 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 453	. . . . . . . . . . . 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 1134	. . . . . . . . . . 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 20	. . . . . . . . . . . . . 14 |- ( ( # ` f ) = 2 -> ( # ` f ) = 2 )
127	126	3ad2ant2 979	. . . . . . . . . . . . 13 |- ( ( f ( V Walks E ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ b = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( # ` f ) = 2 )
128	127	adantl 453	. . . . . . . . . . . 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 453	. . . . . . . . . . 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 695	. . . . . . . . . . . . . . . . 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 27951	. . . . . . . . . . . . . . . . 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 30	. . . . . . . . . . . . . . . 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 2413	. . . . . . . . . . . . . . . . . 18 |- ( b = ( p ` 1 ) -> ( ( 2nd ` ( 1st ` T ) ) = b <-> ( 2nd ` ( 1st ` T ) ) = ( p ` 1 ) ) )
134	133	3anbi2d 1259	. . . . . . . . . . . . . . . . 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 308	. . . . . . . . . . . . . . . 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 211	. . . . . . . . . . . . . . 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 979	. . . . . . . . . . . . . 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 980	. . . . . . . . . . . . 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 420	. . . . . . . . . . . 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 420	. . . . . . . . . . 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 1134	. . . . . . . . . 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 519	. . . . . . . . 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 424	. . . . . . . 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 2790	. . . . . . 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 184	. . . . . 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 245	. . . . 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 1635	. . . 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 2935	. . . . 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 2935	. . . . . 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 1589	. . . . 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 242	. . . 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 279	. . 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 2691	. . 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 253	. 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 245	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 177   /\ wa 359   /\ w3a 936   E.wex 1547   = wceq 1649   e. wcel 1721   E.wrex 2667   {crab 2670   _Vcvv 2916   <.cotp 3778   class class class wbr 4172   X. cxp 4835   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   0cc0 8946   1c1 8947   2c2 10005   #chash 11573   Walks cwalk 21459   WalkOn cwlkon 21463   2WalksOnOt c2wlkonot 28052

This theorem is referenced by:  el2wlkonotot0  28069  el2wlksot  28077  frg2wot1  28160  frg2woteq  28163

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-word 11678  df-wlk 21469  df-wlkon 21475  df-2wlkonot 28055