Theorem 2spthonot3v 25597

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 3736	. . 3 |- ( T e. ( A ( V 2SPathOnOt E ) C ) -> ( A ( V 2SPathOnOt E ) C ) =/= (/) )
2		df-ov 6291	. . . . 5 |- ( A ( V 2SPathOnOt E ) C ) = ( ( V 2SPathOnOt E ) ` <. A , C >. )
3		ndmfv 5887	. . . . 5 |- ( -. <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( ( V 2SPathOnOt E ) ` <. A , C >. ) = (/) )
4	2, 3	syl5eq 2496	. . . 4 |- ( -. <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( A ( V 2SPathOnOt E ) C ) = (/) )
5	4	necon1ai 2650	. . 3 |- ( ( A ( V 2SPathOnOt E ) C ) =/= (/) -> <. A , C >. e. dom ( V 2SPathOnOt E ) )
6		simpl 459	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> v = V )
7		id 22	. . . . . . . . . . . . 13 |- ( v = V -> v = V )
8	7, 7	xpeq12d 4858	. . . . . . . . . . . 12 |- ( v = V -> ( v X. v ) = ( V X. V ) )
9	8, 7	xpeq12d 4858	. . . . . . . . . . 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 6297	. . . . . . . . . . . . . 14 |- ( ( v = V /\ e = E ) -> ( v SPathOn e ) = ( V SPathOn E ) )
12	11	oveqd 6305	. . . . . . . . . . . . 13 |- ( ( v = V /\ e = E ) -> ( a ( v SPathOn e ) b ) = ( a ( V SPathOn E ) b ) )
13	12	breqd 4412	. . . . . . . . . . . 12 |- ( ( v = V /\ e = E ) -> ( f ( a ( v SPathOn e ) b ) p <-> f ( a ( V SPathOn E ) b ) p ) )
14	13	3anbi1d 1342	. . . . . . . . . . 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 1769	. . . . . . . . . 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 3039	. . . . . . . . 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 6350	. . . . . . . 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 25581	. . . . . . . 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 6423	. . . . . . 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 5036	. . . . . 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 2513	. . . . 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 6375	. . . . . . . . 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 3427	. . . . . . . 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 4863	. . . . . . . . . . . 12 |- ( <. A , C >. e. ( V X. V ) <-> ( A e. V /\ C e. V ) )
25		2spthonot 25587	. . . . . . . . . . . . . . 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 2513	. . . . . . . . . . . . . 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 3192	. . . . . . . . . . . . . . 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 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 1187	. . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . 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 219	. . . . . . . . . . . . 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 437	. . . . . . . . . . . 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 199	. . . . . . . . . . 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 32	. . . . . . . . . 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 1027	. . . . . . . . 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 32	. . . . . . . 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 6293	. . . . . . . 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 5035	. . . . . . 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 2546	. . . . . 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 32	. . . . 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 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 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 1001	. . . . 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 985	. . . . . 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 491	. . . . . . . 8 |- ( -. ( V e. _V /\ E e. _V ) <-> ( -. V e. _V \/ -. E e. _V ) )
51	18	mpt2ndm0 6507	. . . . . . . . . . . 12 |- ( -. ( V e. _V /\ E e. _V ) -> ( V 2SPathOnOt E ) = (/) )
52	51	dmeqd 5036	. . . . . . . . . . 11 |- ( -. ( V e. _V /\ E e. _V ) -> dom ( V 2SPathOnOt E ) = dom (/) )
53	52	eleq2d 2513	. . . . . . . . . 10 |- ( -. ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) <-> <. A , C >. e. dom (/) ) )
54		dm0 5047	. . . . . . . . . . 11 |- dom (/) = (/)
55	54	eleq2i 2520	. . . . . . . . . 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 2SPathOnOt E ) <-> <. A , C >. e. (/) ) )
57		noel 3734	. . . . . . . . . 10 |- -. <. A , C >. e. (/)
58	57	pm2.21i 135	. . . . . . . . 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 232	. . . . . . . 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 217	. . . . . . 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 492	. . . . . . . . 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 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 6866	. . . . . . . . . . 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 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 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 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 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 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 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 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 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 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 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 168	. . 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 49	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 370   /\ wa 371   \/ w3o 983   /\ w3a 984   = wceq 1443   E.wex 1662   e. wcel 1886   =/= wne 2621   {crab 2740   _Vcvv 3044   (/)c0 3730   <.cop 3973   class class class wbr 4401   X. cxp 4831   dom cdm 4833   ` cfv 5581  (class class class)co 6288   {coprab 6289   |-> cmpt2 6290   1stc1st 6788   2ndc2nd 6789   1c1 9537   2c2 10656   #chash 12512   SPathOn cspthon 25226   2SPathOnOt c2pthonot 25578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-2spthonot 25581

This theorem is referenced by:  el2spthsoton  25600  2spontn0vne  25608  2spot0  25785