Theorem 2spthonot3v 25683

Description: If an ordered triple represents a simple path of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.)

Assertion

Ref	Expression
2spthonot3v	|- ( T e. ( A ( V 2SPathOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) )

Proof of Theorem 2spthonot3v

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 3728	. . 3 |- ( T e. ( A ( V 2SPathOnOt E ) C ) -> ( A ( V 2SPathOnOt E ) C ) =/= (/) )
2		df-ov 6311	. . . . 5 |- ( A ( V 2SPathOnOt E ) C ) = ( ( V 2SPathOnOt E ) ` <. A , C >. )
3		ndmfv 5903	. . . . 5 |- ( -. <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( ( V 2SPathOnOt E ) ` <. A , C >. ) = (/) )
4	2, 3	syl5eq 2517	. . . 4 |- ( -. <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( A ( V 2SPathOnOt E ) C ) = (/) )
5	4	necon1ai 2670	. . 3 |- ( ( A ( V 2SPathOnOt E ) C ) =/= (/) -> <. A , C >. e. dom ( V 2SPathOnOt E ) )
6		simpl 464	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> v = V )
7		id 22	. . . . . . . . . . . . 13 |- ( v = V -> v = V )
8	7, 7	xpeq12d 4864	. . . . . . . . . . . 12 |- ( v = V -> ( v X. v ) = ( V X. V ) )
9	8, 7	xpeq12d 4864	. . . . . . . . . . 11 |- ( v = V -> ( ( v X. v ) X. v ) = ( ( V X. V ) X. V ) )
10	9	adantr 472	. . . . . . . . . 10 |- ( ( v = V /\ e = E ) -> ( ( v X. v ) X. v ) = ( ( V X. V ) X. V ) )
11		oveq12 6317	. . . . . . . . . . . . . 14 |- ( ( v = V /\ e = E ) -> ( v SPathOn e ) = ( V SPathOn E ) )
12	11	oveqd 6325	. . . . . . . . . . . . 13 |- ( ( v = V /\ e = E ) -> ( a ( v SPathOn e ) b ) = ( a ( V SPathOn E ) b ) )
13	12	breqd 4406	. . . . . . . . . . . 12 |- ( ( v = V /\ e = E ) -> ( f ( a ( v SPathOn e ) b ) p <-> f ( a ( V SPathOn E ) b ) p ) )
14	13	3anbi1d 1369	. . . . . . . . . . 11 |- ( ( v = V /\ e = E ) -> ( ( f ( a ( v SPathOn e ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) <-> ( f ( a ( V SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) ) )
15	14	2exbidv 1778	. . . . . . . . . 10 |- ( ( v = V /\ e = E ) -> ( E. f E. p ( f ( a ( v SPathOn e ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) <-> E. f E. p ( f ( a ( V SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) ) )
16	10, 15	rabeqbidv 3026	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> { t e. ( ( v X. v ) X. v ) | E. f E. p ( f ( a ( v SPathOn 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } )
17	6, 6, 16	mpt2eq123dv 6372	. . . . . . . 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 SPathOn 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) )
18		df-2spthonot 25667	. . . . . . . 8 |- 2SPathOnOt = ( 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 SPathOn e ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) )
19	17, 18	ovmpt2ga 6445	. . . . . . 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( V 2SPathOnOt E ) = ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) )
20	19	dmeqd 5042	. . . . . 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> dom ( V 2SPathOnOt E ) = dom ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) )
21	20	eleq2d 2534	. . . . 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) ) )
22		dmoprabss 6397	. . . . . . . . 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) } C_ ( V X. V )
23	22	sseli 3414	. . . . . . . 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 SPathOn 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 4869	. . . . . . . . . . . 12 |- ( <. A , C >. e. ( V X. V ) <-> ( A e. V /\ C e. V ) )
25		2spthonot 25673	. . . . . . . . . . . . . . 15 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( A ( V 2SPathOnOt E ) C ) = { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( A ( V SPathOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = A /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = C ) ) } )
26	25	eleq2d 2534	. . . . . . . . . . . . . 14 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( T e. ( A ( V 2SPathOnOt E ) C ) <-> T e. { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( A ( V SPathOn E ) C ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = A /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = C ) ) } ) )
27		elrabi 3181	. . . . . . . . . . . . . . 15 |- ( T e. { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( A ( V SPathOn 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 464	. . . . . . . . . . . . . . . . . 18 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( V e. _V /\ E e. _V ) )
29	28	adantr 472	. . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . . 18 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( A e. V /\ C e. V ) )
31	30	adantr 472	. . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . 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 1210	. . . . . . . . . . . . . . . 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 441	. . . . . . . . . . . . . . 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 SPathOn 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 223	. . . . . . . . . . . . 13 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( T e. ( A ( V 2SPathOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) )
37	36	expcom 442	. . . . . . . . . . . 12 |- ( ( A e. V /\ C e. V ) -> ( ( V e. _V /\ E e. _V ) -> ( T e. ( A ( V 2SPathOnOt 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 200	. . . . . . . . . . 11 |- ( <. A , C >. e. ( V X. V ) -> ( ( V e. _V /\ E e. _V ) -> ( T e. ( A ( V 2SPathOnOt 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 2SPathOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
40	39	3adant3 1050	. . . . . . . . 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 SPathOn 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 2SPathOnOt 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( T e. ( A ( V 2SPathOnOt 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 SPathOn 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( T e. ( A ( V 2SPathOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
43		df-mpt2 6313	. . . . . . . 8 |- ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V SPathOn 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) }
44	43	dmeqi 5041	. . . . . . 7 |- dom ( a e. V , b e. V |-> { t e. ( ( V X. V ) X. V ) | E. f E. p ( f ( a ( V SPathOn 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) }
45	42, 44	eleq2s 2567	. . . . . 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 SPathOn 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( T e. ( A ( V 2SPathOnOt 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 SPathOn 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) -> ( T e. ( A ( V 2SPathOnOt 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 223	. . . 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( T e. ( A ( V 2SPathOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
48		3ianor 1024	. . . . 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 SPathOn 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) )
49		df-3or 1008	. . . . . 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 SPathOn 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) )
50		ianor 496	. . . . . . . 8 |- ( -. ( V e. _V /\ E e. _V ) <-> ( -. V e. _V \/ -. E e. _V ) )
51	18	mpt2ndm0 6529	. . . . . . . . . . . 12 |- ( -. ( V e. _V /\ E e. _V ) -> ( V 2SPathOnOt E ) = (/) )
52	51	dmeqd 5042	. . . . . . . . . . 11 |- ( -. ( V e. _V /\ E e. _V ) -> dom ( V 2SPathOnOt E ) = dom (/) )
53	52	eleq2d 2534	. . . . . . . . . 10 |- ( -. ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) <-> <. A , C >. e. dom (/) ) )
54		dm0 5054	. . . . . . . . . . 11 |- dom (/) = (/)
55	54	eleq2i 2541	. . . . . . . . . 10 |- ( <. A , C >. e. dom (/) <-> <. A , C >. e. (/) )
56	53, 55	syl6bb 269	. . . . . . . . 9 |- ( -. ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) <-> <. A , C >. e. (/) ) )
57		noel 3726	. . . . . . . . . 10 |- -. <. A , C >. e. (/)
58	57	pm2.21i 136	. . . . . . . . 9 |- ( <. A , C >. e. (/) -> ( T e. ( A ( V 2SPathOnOt 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 236	. . . . . . . 8 |- ( -. ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( T e. ( A ( V 2SPathOnOt 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 218	. . . . . . 7 |- ( ( -. V e. _V \/ -. E e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( T e. ( A ( V 2SPathOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) )
61		anor 497	. . . . . . . . 9 |- ( ( V e. _V /\ E e. _V ) <-> -. ( -. V e. _V \/ -. E e. _V ) )
62		id 22	. . . . . . . . . . . . 13 |- ( V e. _V -> V e. _V )
63	62	ancri 561	. . . . . . . . . . . 12 |- ( V e. _V -> ( V e. _V /\ V e. _V ) )
64	63	adantr 472	. . . . . . . . . . 11 |- ( ( V e. _V /\ E e. _V ) -> ( V e. _V /\ V e. _V ) )
65		mpt2exga 6888	. . . . . . . . . . 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 SPathOn 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V )
67	66	pm2.24d 139	. . . . . . . . 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( T e. ( A ( V 2SPathOnOt 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 218	. . . . . . . 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( T e. ( A ( V 2SPathOnOt E ) C ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) /\ T e. ( ( V X. V ) X. V ) ) ) ) ) )
69	68	imp 436	. . . . . . 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( T e. ( A ( V 2SPathOnOt 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 980	. . . . . 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( T e. ( A ( V 2SPathOnOt 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 200	. . . . 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( T e. ( A ( V 2SPathOnOt 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 200	. . . 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( T e. ( A ( V 2SPathOnOt 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 169	. . 3 |- ( <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( T e. ( A ( V 2SPathOnOt 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 2SPathOnOt E ) C ) -> ( T e. ( A ( V 2SPathOnOt 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 48	1 |- ( T e. ( A ( V 2SPathOnOt 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 375   /\ wa 376   \/ w3o 1006   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   =/= wne 2641   {crab 2760   _Vcvv 3031   (/)c0 3722   <.cop 3965   class class class wbr 4395   X. cxp 4837   dom cdm 4839   ` cfv 5589  (class class class)co 6308   {coprab 6309   |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811   1c1 9558   2c2 10681   #chash 12553   SPathOn cspthon 25312   2SPathOnOt c2pthonot 25664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-2spthonot 25667

This theorem is referenced by:  el2spthsoton  25686  2spontn0vne  25694  2spot0  25871