Theorem 2spthonot3v 28073

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 3594	. . 3 |- ( T e. ( A ( V 2SPathOnOt E ) C ) -> ( A ( V 2SPathOnOt E ) C ) =/= (/) )
2		df-ov 6043	. . . . 5 |- ( A ( V 2SPathOnOt E ) C ) = ( ( V 2SPathOnOt E ) ` <. A , C >. )
3		ndmfv 5714	. . . . 5 |- ( -. <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( ( V 2SPathOnOt E ) ` <. A , C >. ) = (/) )
4	2, 3	syl5eq 2448	. . . 4 |- ( -. <. A , C >. e. dom ( V 2SPathOnOt E ) -> ( A ( V 2SPathOnOt E ) C ) = (/) )
5	4	necon1ai 2609	. . 3 |- ( ( A ( V 2SPathOnOt E ) C ) =/= (/) -> <. A , C >. e. dom ( V 2SPathOnOt E ) )
6		simpl 444	. . . . . . . . 9 |- ( ( v = V /\ e = E ) -> v = V )
7		id 20	. . . . . . . . . . . . 13 |- ( v = V -> v = V )
8	7, 7	xpeq12d 4862	. . . . . . . . . . . 12 |- ( v = V -> ( v X. v ) = ( V X. V ) )
9	8, 7	xpeq12d 4862	. . . . . . . . . . 11 |- ( v = V -> ( ( v X. v ) X. v ) = ( ( V X. V ) X. V ) )
10	9	adantr 452	. . . . . . . . . 10 |- ( ( v = V /\ e = E ) -> ( ( v X. v ) X. v ) = ( ( V X. V ) X. V ) )
11		oveq12 6049	. . . . . . . . . . . . . 14 |- ( ( v = V /\ e = E ) -> ( v SPathOn e ) = ( V SPathOn E ) )
12	11	oveqd 6057	. . . . . . . . . . . . 13 |- ( ( v = V /\ e = E ) -> ( a ( v SPathOn e ) b ) = ( a ( V SPathOn E ) b ) )
13	12	breqd 4183	. . . . . . . . . . . 12 |- ( ( v = V /\ e = E ) -> ( f ( a ( v SPathOn e ) b ) p <-> f ( a ( V SPathOn E ) b ) p ) )
14	13	3anbi1d 1258	. . . . . . . . . . 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 1635	. . . . . . . . . 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 2911	. . . . . . . . 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 6095	. . . . . . . 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 28057	. . . . . . . 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 6162	. . . . . . 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 5031	. . . . . 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 2471	. . . . 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 6114	. . . . . . . . 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 3304	. . . . . . . 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 4867	. . . . . . . . . . . 12 |- ( <. A , C >. e. ( V X. V ) <-> ( A e. V /\ C e. V ) )
25		2spthonot 28063	. . . . . . . . . . . . . . 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 2471	. . . . . . . . . . . . . 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 3050	. . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . . . . 18 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( V e. _V /\ E e. _V ) )
29	28	adantr 452	. . . . . . . . . . . . . . . . 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 448	. . . . . . . . . . . . . . . . . 18 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( A e. V /\ C e. V ) )
31	30	adantr 452	. . . . . . . . . . . . . . . . 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 448	. . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . 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 424	. . . . . . . . . . . . . . 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 30	. . . . . . . . . . . . . 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 207	. . . . . . . . . . . . 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 425	. . . . . . . . . . . 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 188	. . . . . . . . . . 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 29	. . . . . . . . . 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 977	. . . . . . . . 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 29	. . . . . . . 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 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 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 6045	. . . . . . . 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 5030	. . . . . . 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 2496	. . . . . 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 29	. . . . 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 207	. . . 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 951	. . . . 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 937	. . . . . 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 475	. . . . . . . . 9 |- ( -. ( V e. _V /\ E e. _V ) <-> ( -. V e. _V \/ -. E e. _V ) )
51	18	mpt2ndm0 6432	. . . . . . . . . . . . 13 |- ( -. ( V e. _V /\ E e. _V ) -> ( V 2SPathOnOt E ) = (/) )
52	51	dmeqd 5031	. . . . . . . . . . . 12 |- ( -. ( V e. _V /\ E e. _V ) -> dom ( V 2SPathOnOt E ) = dom (/) )
53	52	eleq2d 2471	. . . . . . . . . . 11 |- ( -. ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) <-> <. A , C >. e. dom (/) ) )
54		dm0 5042	. . . . . . . . . . . 12 |- dom (/) = (/)
55	54	eleq2i 2468	. . . . . . . . . . 11 |- ( <. A , C >. e. dom (/) <-> <. A , C >. e. (/) )
56	53, 55	syl6bb 253	. . . . . . . . . 10 |- ( -. ( V e. _V /\ E e. _V ) -> ( <. A , C >. e. dom ( V 2SPathOnOt E ) <-> <. A , C >. e. (/) ) )
57		noel 3592	. . . . . . . . . . 11 |- -. <. A , C >. e. (/)
58	57	pm2.21i 125	. . . . . . . . . 10 |- ( <. 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 220	. . . . . . . . 9 |- ( -. ( 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 205	. . . . . . . 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 ) ) ) ) )
61		anor 476	. . . . . . . . . 10 |- ( ( V e. _V /\ E e. _V ) <-> -. ( -. V e. _V \/ -. E e. _V ) )
62		id 20	. . . . . . . . . . . . . 14 |- ( V e. _V -> V e. _V )
63	62	ancri 536	. . . . . . . . . . . . 13 |- ( V e. _V -> ( V e. _V /\ V e. _V ) )
64	63	adantr 452	. . . . . . . . . . . 12 |- ( ( V e. _V /\ E e. _V ) -> ( V e. _V /\ V e. _V ) )
65		mpt2exga 6383	. . . . . . . . . . . 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 SPathOn 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 SPathOn E ) b ) p /\ ( # ` f ) = 2 /\ ( ( 1st ` ( 1st ` t ) ) = a /\ ( 2nd ` ( 1st ` t ) ) = ( p ` 1 ) /\ ( 2nd ` t ) = b ) ) } ) e. _V )
67	66	pm2.24d 137	. . . . . . . . . 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 -> ( <. 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 205	. . . . . . . . 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 ) ) ) ) ) )
69	68	imp 419	. . . . . . . 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 ) ) ) ) )
70	60, 69	jaoi 369	. . . . . . 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 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	70	jaoi2 934	. . . . . 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 ) ) ) ) )
72	49, 71	sylbi 188	. . . . 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 ) ) ) ) )
73	48, 72	sylbi 188	. . . 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 ) ) ) ) )
74	47, 73	pm2.61i 158	. . 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 ) ) ) )
75	1, 5, 74	3syl 19	. 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 ) ) ) )
76	75	pm2.43i 45	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 358   /\ wa 359   \/ w3o 935   /\ w3a 936   E.wex 1547   = wceq 1649   e. wcel 1721   =/= wne 2567   {crab 2670   _Vcvv 2916   (/)c0 3588   <.cop 3777   class class class wbr 4172   X. cxp 4835   dom cdm 4837   ` cfv 5413  (class class class)co 6040   {coprab 6041   e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   1c1 8947   2c2 10005   #chash 11573   SPathOn cspthon 21466   2SPathOnOt c2pthonot 28054

This theorem is referenced by:  el2spthsoton  28076  2spontn0vne  28084  2spot0  28167

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

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-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-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  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-2spthonot 28057