Theorem 2wlkonot3v 30413

Description: If an ordered triple represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)

Assertion

Ref	Expression
2wlkonot3v	|- ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) )

Proof of Theorem 2wlkonot3v

Dummy variables f p t a b c e v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ne0i 3658	. . 3 |- ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( A ( V 2WalksOnOt E ) C ) =/= (/) )
2		df-ov 6109	. . . . 5 |- ( A ( V 2WalksOnOt E ) C ) = ( ( V 2WalksOnOt E ) ` <. A , C >. )
3		ndmfv 5729	. . . . 5 |- ( -. <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( ( V 2WalksOnOt E ) ` <. A , C >. ) = (/) )
4	2, 3	syl5eq 2487	. . . 4 |- ( -. <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( A ( V 2WalksOnOt E ) C ) = (/) )
5	4	necon1ai 2668	. . 3 |- ( ( A ( V 2WalksOnOt E ) C ) =/= (/) -> <. A , C >. e. dom ( V 2WalksOnOt E ) )
6		simpl 457	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> v = V )
7		id 22	. . . . . . . . . . . . 13 |- ( v = V -> v = V )
8	7, 7	xpeq12d 4880	. . . . . . . . . . . 12 |- ( v = V -> ( v X. v ) = ( V X. V ) )
9	8, 7	xpeq12d 4880	. . . . . . . . . . 11 |- ( v = V -> ( ( v X. v ) X. v ) = ( ( V X. V ) X. V ) )
10	9	adantr 465	. . . . . . . . . 10 |- ( ( v = V /\ e = E ) -> ( ( v X. v ) X. v ) = ( ( V X. V ) X. V ) )
11		oveq12 6115	. . . . . . . . . . . . . 14 |- ( ( v = V /\ e = E ) -> ( v WalkOn e ) = ( V WalkOn E ) )
12	11	oveqd 6123	. . . . . . . . . . . . 13 |- ( ( v = V /\ e = E ) -> ( a ( v WalkOn e ) b ) = ( a ( V WalkOn E ) b ) )
13	12	breqd 4318	. . . . . . . . . . . 12 |- ( ( v = V /\ e = E ) -> ( f ( a ( v WalkOn e ) b ) p <-> f ( a ( V WalkOn E ) b ) p ) )
14	13	3anbi1d 1293	. . . . . . . . . . 11 |- ( ( v = V /\ e = E ) -> ( ( f ( a ( v WalkOn e ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) <-> ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) ) )
15	14	2exbidv 1682	. . . . . . . . . 10 |- ( ( v = V /\ e = E ) -> ( E. f E. p ( f ( a ( v WalkOn e ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) <-> E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) ) )
16	10, 15	rabeqbidv 2982	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> { t e. ( ( v X. v ) X. v ) | E. f E. p ( f ( a ( v WalkOn e ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } = { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } )
17	6, 6, 16	mpt2eq123dv 6163	. . . . . . . 8 |- ( ( v = V /\ e = E ) -> ( a e. v , b e. v |-> { t e. ( ( v X. v ) X. v ) | E. f E. p ( f ( a ( v WalkOn e ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) = ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) )
18		df-2wlkonot 30396	. . . . . . . 8 |- 2WalksOnOt = ( v e. _V , e e. _V |-> ( a e. v , b e. v |-> { t e. ( ( v X. v ) X. v ) | E. f E. p ( f ( a ( v WalkOn e ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) )
19	17, 18	ovmpt2ga 6235	. . . . . . 7 |- ( ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( V 2WalksOnOt E ) = ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) )
20	19	dmeqd 5057	. . . . . 6 |- ( ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> dom ( V 2WalksOnOt E ) = dom ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) )
21	20	eleq2d 2510	. . . . 5 |- ( ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) <-> <. A , C >. e. dom ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) ) )
22		dmoprabss 6187	. . . . . . . . 9 |- dom { <. <. a , b >. , c >. | ( ( a e. V /\ b e. V ) /\ c = { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) } C_ ( V X. V )
23	22	sseli 3367	. . . . . . . 8 |- ( <. A , C >. e. dom { <. <. a , b >. , c >. | ( ( a e. V /\ b e. V ) /\ c = { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) } -> <. A , C >. e. ( V X. V ) )
24		opelxp 4884	. . . . . . . . . . . 12 |- ( <. A , C >. e. ( V X. V ) <-> ( A e. V /\ C e. V ) )
25		2wlkonot 30403	. . . . . . . . . . . . . . 15 |- ( ( ( V e. _V /\ E e. _V ) /\ ( 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 ) ) } )
26	25	eleq2d 2510	. . . . . . . . . . . . . 14 |- ( ( ( V e. _V /\ E e. _V ) /\ ( 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 ) ) } ) )
27		elrabi 3129	. . . . . . . . . . . . . . 15 |- ( 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 ) )
28		simpl 457	. . . . . . . . . . . . . . . . . 18 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( V e. _V /\ E e. _V ) )
29	28	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) /\ T e. ( ( V X. V ) X. V ) ) -> ( V e. _V /\ E e. _V ) )
30		simpr 461	. . . . . . . . . . . . . . . . . 18 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( A e. V /\ C e. V ) )
31	30	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) /\ T e. ( ( V X. V ) X. V ) ) -> ( A e. V /\ C e. V ) )
32		simpr 461	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) /\ T e. ( ( V X. V ) X. V ) ) -> T e. ( ( V X. V ) X. V ) )
33	29, 31, 32	3jca 1168	. . . . . . . . . . . . . . . 16 |- ( ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) /\ T e. ( ( V X. V ) X. V ) ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) )
34	33	ex 434	. . . . . . . . . . . . . . 15 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( T e. ( ( V X. V ) X. V ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) )
35	27, 34	syl5 32	. . . . . . . . . . . . . 14 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( 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 ) ) } -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) )
36	26, 35	sylbid 215	. . . . . . . . . . . . 13 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) )
37	36	expcom 435	. . . . . . . . . . . 12 |- ( ( A e. V /\ C e. V ) -> ( ( V e. _V /\ E e. _V ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
38	24, 37	sylbi 195	. . . . . . . . . . 11 |- ( <. A , C >. e. ( V X. V ) -> ( ( V e. _V /\ E e. _V ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
39	38	com12 31	. . . . . . . . . 10 |- ( ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. ( V X. V ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
40	39	3adant3 1008	. . . . . . . . 9 |- ( ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. ( V X. V ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
41	40	com12 31	. . . . . . . 8 |- ( <. A , C >. e. ( V X. V ) -> ( ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
42	23, 41	syl 16	. . . . . . 7 |- ( <. A , C >. e. dom { <. <. a , b >. , c >. | ( ( a e. V /\ b e. V ) /\ c = { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) } -> ( ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
43		df-mpt2 6111	. . . . . . . 8 |- ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) = { <. <. a , b >. , c >. | ( ( a e. V /\ b e. V ) /\ c = { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) }
44	43	dmeqi 5056	. . . . . . 7 |- dom ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) = dom { <. <. a , b >. , c >. | ( ( a e. V /\ b e. V ) /\ c = { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) }
45	42, 44	eleq2s 2535	. . . . . 6 |- ( <. A , C >. e. dom ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) -> ( ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
46	45	com12 31	. . . . 5 |- ( ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
47	21, 46	sylbid 215	. . . 4 |- ( ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
48		3ianor 982	. . . . 5 |- ( -. ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) <-> ( -. V e. _V \/ -. E e. _V \/ -. ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) )
49		df-3or 966	. . . . . 6 |- ( ( -. V e. _V \/ -. E e. _V \/ -. ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) <-> ( ( -. V e. _V \/ -. E e. _V ) \/ -. ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) )
50		ianor 488	. . . . . . . . 9 |- ( -. ( V e. _V /\ E e. _V ) <-> ( -. V e. _V \/ -. E e. _V ) )
51	18	mpt2ndm0 6754	. . . . . . . . . . . . 13 |- ( -. ( V e. _V /\ E e. _V ) -> ( V 2WalksOnOt E ) = (/) )
52	51	dmeqd 5057	. . . . . . . . . . . 12 |- ( -. ( V e. _V /\ E e. _V ) -> dom ( V 2WalksOnOt E ) = dom (/) )
53	52	eleq2d 2510	. . . . . . . . . . 11 |- ( -. ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) <-> <. A , C >. e. dom (/) ) )
54		dm0 5068	. . . . . . . . . . . 12 |- dom (/) = (/)
55	54	eleq2i 2507	. . . . . . . . . . 11 |- ( <. A , C >. e. dom (/) <-> <. A , C >. e. (/) )
56	53, 55	syl6bb 261	. . . . . . . . . 10 |- ( -. ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) <-> <. A , C >. e. (/) ) )
57		noel 3656	. . . . . . . . . . 11 |- -. <. A , C >. e. (/)
58	57	pm2.21i 131	. . . . . . . . . 10 |- ( <. A , C >. e. (/) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) )
59	56, 58	syl6bi 228	. . . . . . . . 9 |- ( -. ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
60	50, 59	sylbir 213	. . . . . . . 8 |- ( ( -. V e. _V \/ -. E e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
61		anor 489	. . . . . . . . . 10 |- ( ( V e. _V /\ E e. _V ) <-> -. ( -. V e. _V \/ -. E e. _V ) )
62		id 22	. . . . . . . . . . . . . 14 |- ( V e. _V -> V e. _V )
63	62	ancri 552	. . . . . . . . . . . . 13 |- ( V e. _V -> ( V e. _V /\ V e. _V ) )
64	63	adantr 465	. . . . . . . . . . . 12 |- ( ( V e. _V /\ E e. _V ) -> ( V e. _V /\ V e. _V ) )
65		mpt2exga 6664	. . . . . . . . . . . 12 |- ( ( V e. _V /\ V e. _V ) -> ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V )
66	64, 65	syl 16	. . . . . . . . . . 11 |- ( ( V e. _V /\ E e. _V ) -> ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V )
67	66	pm2.24d 143	. . . . . . . . . 10 |- ( ( V e. _V /\ E e. _V ) -> ( -. ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) ) )
68	61, 67	sylbir 213	. . . . . . . . 9 |- ( -. ( -. V e. _V \/ -. E e. _V ) -> ( -. ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) ) )
69	68	imp 429	. . . . . . . 8 |- ( ( -. ( -. V e. _V \/ -. E e. _V ) /\ -. ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
70	60, 69	jaoi 379	. . . . . . 7 |- ( ( ( -. V e. _V \/ -. E e. _V ) \/ ( -. ( -. V e. _V \/ -. E e. _V ) /\ -. ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
71	70	jaoi2 959	. . . . . 6 |- ( ( ( -. V e. _V \/ -. E e. _V ) \/ -. ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
72	49, 71	sylbi 195	. . . . 5 |- ( ( -. V e. _V \/ -. E e. _V \/ -. ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
73	48, 72	sylbi 195	. . . 4 |- ( -. ( V e. _V /\ E e. _V /\ ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V WalkOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
74	47, 73	pm2.61i 164	. . 3 |- ( <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) )
75	1, 5, 74	3syl 20	. 2 |- ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) )
76	75	pm2.43i 47	1 |- ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   \/ w3o 964   /\ w3a 965   = wceq 1369   E.wex 1586   e. wcel 1756   =/= wne 2620   {crab 2734   _Vcvv 2987   (/)c0 3652   <.cop 3898   class class class wbr 4307   X. cxp 4853   dom cdm 4855   ` cfv 5433  (class class class)co 6106   {coprab 6107   e. cmpt2 6108   1stc1st 6590   2ndc2nd 6591   1c1 9298   2c2 10386   #chash 12118   WalkOn cwlkon 23424   2WalksOnOt c2wlkonot 30393

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 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387

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 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-1st 6592  df-2nd 6593  df-2wlkonot 30396

This theorem is referenced by:  2wlkonotv  30415  el2wlksoton  30416  frg2woteq  30672