Theorem 2wlkonot3v 25595

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 3768	. . 3 |- ( T e. ( A ( V 2WalksOnOt E ) C ) -> ( A ( V 2WalksOnOt E ) C ) =/= (/) )
2		df-ov 6306	. . . . 5 |- ( A ( V 2WalksOnOt E ) C ) = ( ( V 2WalksOnOt E ) ` <. A , C >. )
3		ndmfv 5903	. . . . 5 |- ( -. <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( ( V 2WalksOnOt E ) ` <. A , C >. ) = (/) )
4	2, 3	syl5eq 2476	. . . 4 |- ( -. <. A , C >. e. dom ( V 2WalksOnOt E ) -> ( A ( V 2WalksOnOt E ) C ) = (/) )
5	4	necon1ai 2656	. . 3 |- ( ( A ( V 2WalksOnOt E ) C ) =/= (/) -> <. A , C >. e. dom ( V 2WalksOnOt E ) )
6		simpl 459	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> v = V )
7		id 23	. . . . . . . . . . . . 13 |- ( v = V -> v = V )
8	7, 7	xpeq12d 4876	. . . . . . . . . . . 12 |- ( v = V -> ( v X. v ) = ( V X. V ) )
9	8, 7	xpeq12d 4876	. . . . . . . . . . 11 |- ( v = V -> ( ( v X. v ) X. v ) = ( ( V X. V ) X. V ) )
10	9	adantr 467	. . . . . . . . . 10 |- ( ( v = V /\ e = E ) -> ( ( v X. v ) X. v ) = ( ( V X. V ) X. V ) )
11		oveq12 6312	. . . . . . . . . . . . . 14 |- ( ( v = V /\ e = E ) -> ( v WalkOn e ) = ( V WalkOn E ) )
12	11	oveqd 6320	. . . . . . . . . . . . 13 |- ( ( v = V /\ e = E ) -> ( a ( v WalkOn e ) b ) = ( a ( V WalkOn E ) b ) )
13	12	breqd 4432	. . . . . . . . . . . 12 |- ( ( v = V /\ e = E ) -> ( f ( a ( v WalkOn e ) b ) p <-> f ( a ( V WalkOn E ) b ) p ) )
14	13	3anbi1d 1340	. . . . . . . . . . 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 1761	. . . . . . . . . 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 3077	. . . . . . . . 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 6365	. . . . . . . 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 25578	. . . . . . . 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 6438	. . . . . . 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 5054	. . . . . 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 2493	. . . . 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 6390	. . . . . . . . 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 3461	. . . . . . . 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 4881	. . . . . . . . . . . 12 |- ( <. A , C >. e. ( V X. V ) <-> ( A e. V /\ C e. V ) )
25		2wlkonot 25585	. . . . . . . . . . . . . . 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 2493	. . . . . . . . . . . . . 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 3227	. . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . 18 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( V e. _V /\ E e. _V ) )
29	28	adantr 467	. . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . . 18 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( A e. V /\ C e. V ) )
31	30	adantr 467	. . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . 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 1186	. . . . . . . . . . . . . . . 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 436	. . . . . . . . . . . . . . 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 34	. . . . . . . . . . . . . 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 219	. . . . . . . . . . . . 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 437	. . . . . . . . . . . 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 199	. . . . . . . . . . 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 33	. . . . . . . . . 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 1026	. . . . . . . . 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 33	. . . . . . . 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 17	. . . . . . 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 6308	. . . . . . . 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 5053	. . . . . . 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 2531	. . . . . 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 33	. . . . 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 219	. . . 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 1000	. . . . 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 984	. . . . . 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 491	. . . . . . . 8 |- ( -. ( V e. _V /\ E e. _V ) <-> ( -. V e. _V \/ -. E e. _V ) )
51	18	mpt2ndm0 6522	. . . . . . . . . . . 12 |- ( -. ( V e. _V /\ E e. _V ) -> ( V 2WalksOnOt E ) = (/) )
52	51	dmeqd 5054	. . . . . . . . . . 11 |- ( -. ( V e. _V /\ E e. _V ) -> dom ( V 2WalksOnOt E ) = dom (/) )
53	52	eleq2d 2493	. . . . . . . . . 10 |- ( -. ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) <-> <. A , C >. e. dom (/) ) )
54		dm0 5065	. . . . . . . . . . 11 |- dom (/) = (/)
55	54	eleq2i 2501	. . . . . . . . . 10 |- ( <. A , C >. e. dom (/) <-> <. A , C >. e. (/) )
56	53, 55	syl6bb 265	. . . . . . . . 9 |- ( -. ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. dom ( V 2WalksOnOt E ) <-> <. A , C >. e. (/) ) )
57		noel 3766	. . . . . . . . . 10 |- -. <. A , C >. e. (/)
58	57	pm2.21i 135	. . . . . . . . 9 |- ( <. 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 232	. . . . . . . 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 ) ) ) ) )
60	50, 59	sylbir 217	. . . . . . 7 |- ( ( -. 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 492	. . . . . . . . 9 |- ( ( V e. _V /\ E e. _V ) <-> -. ( -. V e. _V \/ -. E e. _V ) )
62		id 23	. . . . . . . . . . . . 13 |- ( V e. _V -> V e. _V )
63	62	ancri 555	. . . . . . . . . . . 12 |- ( V e. _V -> ( V e. _V /\ V e. _V ) )
64	63	adantr 467	. . . . . . . . . . 11 |- ( ( V e. _V /\ E e. _V ) -> ( V e. _V /\ V e. _V ) )
65		mpt2exga 6881	. . . . . . . . . . 11 |- ( ( 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 17	. . . . . . . . . 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 )
67	66	pm2.24d 138	. . . . . . . . 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 ) ) ) ) ) )
68	61, 67	sylbir 217	. . . . . . . 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 ) ) ) ) ) )
69	68	imp 431	. . . . . . 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 ) -> ( <. 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	jaoi3 979	. . . . . 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 ) ) ) ) )
71	49, 70	sylbi 199	. . . . 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 ) ) ) ) )
72	48, 71	sylbi 199	. . . 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 ) ) ) ) )
73	47, 72	pm2.61i 168	. . 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 ) ) ) )
74	1, 5, 73	3syl 18	. 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 ) ) ) )
75	74	pm2.43i 50	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 370   /\ wa 371   \/ w3o 982   /\ w3a 983   = wceq 1438   E.wex 1660   e. wcel 1869   =/= wne 2619   {crab 2780   _Vcvv 3082   (/)c0 3762   <.cop 4003   class class class wbr 4421   X. cxp 4849   dom cdm 4851   ` cfv 5599  (class class class)co 6303   {coprab 6304   |-> cmpt2 6305   1stc1st 6803   2ndc2nd 6804   1c1 9542   2c2 10661   #chash 12516   WalkOn cwlkon 25222   2WalksOnOt c2wlkonot 25575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-2wlkonot 25578

This theorem is referenced by:  2wlkonotv  25597  el2wlksoton  25598  frg2woteq  25780