Theorem 2pthon3v 26839

Description: For a vertex adjacent to two other vertices there is a simple path of length 2 between these other vertices in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.)

Hypotheses

Ref	Expression
2pthon3v.v	|- V = (Vtx ` G )
2pthon3v.e	|- E = (Edg ` G )

Assertion

Ref	Expression
2pthon3v	|- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( { A , B } e. E /\ { B , C } e. E ) ) -> E. f E. p ( f ( A (SPathsOn ` G ) C ) p /\ ( # ` f ) = 2 ) )

Distinct variable groups:   A, f, p   B, f, p   C, f, p   f, G, p

Allowed substitution hints:   E( f, p)   V( f, p)

Proof of Theorem 2pthon3v

Dummy variables i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2pthon3v.e	. . . . . . . . . 10 |- E = (Edg ` G )
2		edgval 25941	. . . . . . . . . 10 |- (Edg ` G ) = ran (iEdg ` G )
3	1, 2	eqtri 2644	. . . . . . . . 9 |- E = ran (iEdg ` G )
4	3	eleq2i 2693	. . . . . . . 8 |- ( { A , B } e. E <-> { A , B } e. ran (iEdg ` G ) )
5		2pthon3v.v	. . . . . . . . . . 11 |- V = (Vtx ` G )
6		eqid 2622	. . . . . . . . . . 11 |- (iEdg ` G ) = (iEdg ` G )
7	5, 6	uhgrf 25957	. . . . . . . . . 10 |- ( G e. UHGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> ( ~P V \ { (/) } ) )
8	7	ffnd 6046	. . . . . . . . 9 |- ( G e. UHGraph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
9		fvelrnb 6243	. . . . . . . . 9 |- ( (iEdg ` G ) Fn dom (iEdg ` G ) -> ( { A , B } e. ran (iEdg ` G ) <-> E. i e. dom (iEdg ` G ) ( (iEdg ` G ) ` i ) = { A , B } ) )
10	8, 9	syl 17	. . . . . . . 8 |- ( G e. UHGraph -> ( { A , B } e. ran (iEdg ` G ) <-> E. i e. dom (iEdg ` G ) ( (iEdg ` G ) ` i ) = { A , B } ) )
11	4, 10	syl5bb 272	. . . . . . 7 |- ( G e. UHGraph -> ( { A , B } e. E <-> E. i e. dom (iEdg ` G ) ( (iEdg ` G ) ` i ) = { A , B } ) )
12	3	eleq2i 2693	. . . . . . . 8 |- ( { B , C } e. E <-> { B , C } e. ran (iEdg ` G ) )
13		fvelrnb 6243	. . . . . . . . 9 |- ( (iEdg ` G ) Fn dom (iEdg ` G ) -> ( { B , C } e. ran (iEdg ` G ) <-> E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = { B , C } ) )
14	8, 13	syl 17	. . . . . . . 8 |- ( G e. UHGraph -> ( { B , C } e. ran (iEdg ` G ) <-> E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = { B , C } ) )
15	12, 14	syl5bb 272	. . . . . . 7 |- ( G e. UHGraph -> ( { B , C } e. E <-> E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = { B , C } ) )
16	11, 15	anbi12d 747	. . . . . 6 |- ( G e. UHGraph -> ( ( { A , B } e. E /\ { B , C } e. E ) <-> ( E. i e. dom (iEdg ` G ) ( (iEdg ` G ) ` i ) = { A , B } /\ E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = { B , C } ) ) )
17	16	adantr 481	. . . . 5 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( { A , B } e. E /\ { B , C } e. E ) <-> ( E. i e. dom (iEdg ` G ) ( (iEdg ` G ) ` i ) = { A , B } /\ E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = { B , C } ) ) )
18	17	adantr 481	. . . 4 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( { A , B } e. E /\ { B , C } e. E ) <-> ( E. i e. dom (iEdg ` G ) ( (iEdg ` G ) ` i ) = { A , B } /\ E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = { B , C } ) ) )
19		reeanv 3107	. . . 4 |- ( E. i e. dom (iEdg ` G ) E. j e. dom (iEdg ` G ) ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) <-> ( E. i e. dom (iEdg ` G ) ( (iEdg ` G ) ` i ) = { A , B } /\ E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = { B , C } ) )
20	18, 19	syl6bbr 278	. . 3 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( { A , B } e. E /\ { B , C } e. E ) <-> E. i e. dom (iEdg ` G ) E. j e. dom (iEdg ` G ) ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) ) )
21		df-s2 13593	. . . . . . . 8 |- <" i j "> = ( <" i "> ++ <" j "> )
22	21	ovexi 6679	. . . . . . 7 |- <" i j "> e. _V
23		df-s3 13594	. . . . . . . 8 |- <" A B C "> = ( <" A B "> ++ <" C "> )
24	23	ovexi 6679	. . . . . . 7 |- <" A B C "> e. _V
25	22, 24	pm3.2i 471	. . . . . 6 |- ( <" i j "> e. _V /\ <" A B C "> e. _V )
26		eqid 2622	. . . . . . . 8 |- <" A B C "> = <" A B C ">
27		eqid 2622	. . . . . . . 8 |- <" i j "> = <" i j ">
28		simp-4r 807	. . . . . . . 8 |- ( ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i e. dom (iEdg ` G ) /\ j e. dom (iEdg ` G ) ) ) /\ ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) ) -> ( A e. V /\ B e. V /\ C e. V ) )
29		3simpb 1059	. . . . . . . . 9 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( A =/= B /\ B =/= C ) )
30	29	ad3antlr 767	. . . . . . . 8 |- ( ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i e. dom (iEdg ` G ) /\ j e. dom (iEdg ` G ) ) ) /\ ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) ) -> ( A =/= B /\ B =/= C ) )
31		eqimss2 3658	. . . . . . . . . 10 |- ( ( (iEdg ` G ) ` i ) = { A , B } -> { A , B } C_ ( (iEdg ` G ) ` i ) )
32		eqimss2 3658	. . . . . . . . . 10 |- ( ( (iEdg ` G ) ` j ) = { B , C } -> { B , C } C_ ( (iEdg ` G ) ` j ) )
33	31, 32	anim12i 590	. . . . . . . . 9 |- ( ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) -> ( { A , B } C_ ( (iEdg ` G ) ` i ) /\ { B , C } C_ ( (iEdg ` G ) ` j ) ) )
34	33	adantl 482	. . . . . . . 8 |- ( ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i e. dom (iEdg ` G ) /\ j e. dom (iEdg ` G ) ) ) /\ ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) ) -> ( { A , B } C_ ( (iEdg ` G ) ` i ) /\ { B , C } C_ ( (iEdg ` G ) ` j ) ) )
35		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( i = j -> ( (iEdg ` G ) ` i ) = ( (iEdg ` G ) ` j ) )
36	35	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( i = j -> ( ( (iEdg ` G ) ` i ) = { A , B } <-> ( (iEdg ` G ) ` j ) = { A , B } ) )
37	36	anbi1d 741	. . . . . . . . . . . . 13 |- ( i = j -> ( ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) <-> ( ( (iEdg ` G ) ` j ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) ) )
38		eqtr2 2642	. . . . . . . . . . . . . 14 |- ( ( ( (iEdg ` G ) ` j ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) -> { A , B } = { B , C } )
39		3simpa 1058	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( A e. V /\ B e. V ) )
40		3simpc 1060	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( B e. V /\ C e. V ) )
41		preq12bg 4386	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( A e. V /\ B e. V ) /\ ( B e. V /\ C e. V ) ) -> ( { A , B } = { B , C } <-> ( ( A = B /\ B = C ) \/ ( A = C /\ B = B ) ) ) )
42	39, 40, 41	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( { A , B } = { B , C } <-> ( ( A = B /\ B = C ) \/ ( A = C /\ B = B ) ) ) )
43		eqneqall 2805	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( A = B -> ( A =/= B -> i =/= j ) )
44	43	com12 32	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( A =/= B -> ( A = B -> i =/= j ) )
45	44	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( A = B -> i =/= j ) )
46	45	com12 32	. . . . . . . . . . . . . . . . . . . . 21 |- ( A = B -> ( ( A =/= B /\ A =/= C /\ B =/= C ) -> i =/= j ) )
47	46	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A = B /\ B = C ) -> ( ( A =/= B /\ A =/= C /\ B =/= C ) -> i =/= j ) )
48		eqneqall 2805	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( A = C -> ( A =/= C -> i =/= j ) )
49	48	com12 32	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( A =/= C -> ( A = C -> i =/= j ) )
50	49	3ad2ant2 1083	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( A = C -> i =/= j ) )
51	50	com12 32	. . . . . . . . . . . . . . . . . . . . 21 |- ( A = C -> ( ( A =/= B /\ A =/= C /\ B =/= C ) -> i =/= j ) )
52	51	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A = C /\ B = B ) -> ( ( A =/= B /\ A =/= C /\ B =/= C ) -> i =/= j ) )
53	47, 52	jaoi 394	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A = B /\ B = C ) \/ ( A = C /\ B = B ) ) -> ( ( A =/= B /\ A =/= C /\ B =/= C ) -> i =/= j ) )
54	42, 53	syl6bi 243	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( { A , B } = { B , C } -> ( ( A =/= B /\ A =/= C /\ B =/= C ) -> i =/= j ) ) )
55	54	com23 86	. . . . . . . . . . . . . . . . 17 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B } = { B , C } -> i =/= j ) ) )
56	55	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B } = { B , C } -> i =/= j ) ) )
57	56	imp 445	. . . . . . . . . . . . . . 15 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( { A , B } = { B , C } -> i =/= j ) )
58	57	com12 32	. . . . . . . . . . . . . 14 |- ( { A , B } = { B , C } -> ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> i =/= j ) )
59	38, 58	syl 17	. . . . . . . . . . . . 13 |- ( ( ( (iEdg ` G ) ` j ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) -> ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> i =/= j ) )
60	37, 59	syl6bi 243	. . . . . . . . . . . 12 |- ( i = j -> ( ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) -> ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> i =/= j ) ) )
61	60	com23 86	. . . . . . . . . . 11 |- ( i = j -> ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) -> i =/= j ) ) )
62		2a1 28	. . . . . . . . . . 11 |- ( i =/= j -> ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) -> i =/= j ) ) )
63	61, 62	pm2.61ine 2877	. . . . . . . . . 10 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) -> i =/= j ) )
64	63	adantr 481	. . . . . . . . 9 |- ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i e. dom (iEdg ` G ) /\ j e. dom (iEdg ` G ) ) ) -> ( ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) -> i =/= j ) )
65	64	imp 445	. . . . . . . 8 |- ( ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i e. dom (iEdg ` G ) /\ j e. dom (iEdg ` G ) ) ) /\ ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) ) -> i =/= j )
66		simplr2 1104	. . . . . . . . 9 |- ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i e. dom (iEdg ` G ) /\ j e. dom (iEdg ` G ) ) ) -> A =/= C )
67	66	adantr 481	. . . . . . . 8 |- ( ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i e. dom (iEdg ` G ) /\ j e. dom (iEdg ` G ) ) ) /\ ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) ) -> A =/= C )
68	26, 27, 28, 30, 34, 5, 6, 65, 67	2pthond 26838	. . . . . . 7 |- ( ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i e. dom (iEdg ` G ) /\ j e. dom (iEdg ` G ) ) ) /\ ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) ) -> <" i j "> ( A (SPathsOn ` G ) C ) <" A B C "> )
69		s2len 13634	. . . . . . 7 |- ( # ` <" i j "> ) = 2
70	68, 69	jctir 561	. . . . . 6 |- ( ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i e. dom (iEdg ` G ) /\ j e. dom (iEdg ` G ) ) ) /\ ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) ) -> ( <" i j "> ( A (SPathsOn ` G ) C ) <" A B C "> /\ ( # ` <" i j "> ) = 2 ) )
71		breq12 4658	. . . . . . . 8 |- ( ( f = <" i j "> /\ p = <" A B C "> ) -> ( f ( A (SPathsOn ` G ) C ) p <-> <" i j "> ( A (SPathsOn ` G ) C ) <" A B C "> ) )
72		fveq2 6191	. . . . . . . . . 10 |- ( f = <" i j "> -> ( # ` f ) = ( # ` <" i j "> ) )
73	72	eqeq1d 2624	. . . . . . . . 9 |- ( f = <" i j "> -> ( ( # ` f ) = 2 <-> ( # ` <" i j "> ) = 2 ) )
74	73	adantr 481	. . . . . . . 8 |- ( ( f = <" i j "> /\ p = <" A B C "> ) -> ( ( # ` f ) = 2 <-> ( # ` <" i j "> ) = 2 ) )
75	71, 74	anbi12d 747	. . . . . . 7 |- ( ( f = <" i j "> /\ p = <" A B C "> ) -> ( ( f ( A (SPathsOn ` G ) C ) p /\ ( # ` f ) = 2 ) <-> ( <" i j "> ( A (SPathsOn ` G ) C ) <" A B C "> /\ ( # ` <" i j "> ) = 2 ) ) )
76	75	spc2egv 3295	. . . . . 6 |- ( ( <" i j "> e. _V /\ <" A B C "> e. _V ) -> ( ( <" i j "> ( A (SPathsOn ` G ) C ) <" A B C "> /\ ( # ` <" i j "> ) = 2 ) -> E. f E. p ( f ( A (SPathsOn ` G ) C ) p /\ ( # ` f ) = 2 ) ) )
77	25, 70, 76	mpsyl 68	. . . . 5 |- ( ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i e. dom (iEdg ` G ) /\ j e. dom (iEdg ` G ) ) ) /\ ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) ) -> E. f E. p ( f ( A (SPathsOn ` G ) C ) p /\ ( # ` f ) = 2 ) )
78	77	ex 450	. . . 4 |- ( ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( i e. dom (iEdg ` G ) /\ j e. dom (iEdg ` G ) ) ) -> ( ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) -> E. f E. p ( f ( A (SPathsOn ` G ) C ) p /\ ( # ` f ) = 2 ) ) )
79	78	rexlimdvva 3038	. . 3 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( E. i e. dom (iEdg ` G ) E. j e. dom (iEdg ` G ) ( ( (iEdg ` G ) ` i ) = { A , B } /\ ( (iEdg ` G ) ` j ) = { B , C } ) -> E. f E. p ( f ( A (SPathsOn ` G ) C ) p /\ ( # ` f ) = 2 ) ) )
80	20, 79	sylbid 230	. 2 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( { A , B } e. E /\ { B , C } e. E ) -> E. f E. p ( f ( A (SPathsOn ` G ) C ) p /\ ( # ` f ) = 2 ) ) )
81	80	3impia 1261	1 |- ( ( ( G e. UHGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) /\ ( { A , B } e. E /\ { B , C } e. E ) ) -> E. f E. p ( f ( A (SPathsOn ` G ) C ) p /\ ( # ` f ) = 2 ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179   class class class wbr 4653   dom cdm 5114   ran crn 5115   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   2c2 11070   #chash 13117   ++ cconcat 13293   <"cs1 13294   <"cs2 13586   <"cs3 13587  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951  SPathsOncspthson 26611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-edg 25940  df-uhgr 25953  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-spthson 26615

This theorem is referenced by:  2pthfrgr  27148