Theorem uhgr3cyclex 27042

Description: If there are three different vertices in a hypergraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)

Hypotheses

Ref	Expression
uhgr3cyclex.v	|- V = (Vtx ` G )
uhgr3cyclex.e	|- E = (Edg ` G )

Assertion

Ref	Expression
uhgr3cyclex	|- ( ( 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 /\ { C , A } e. E ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) )

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 uhgr3cyclex

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

Step	Hyp	Ref	Expression
1		uhgr3cyclex.e	. . . . . . 7 |- E = (Edg ` G )
2	1	eleq2i 2693	. . . . . 6 |- ( { A , B } e. E <-> { A , B } e. (Edg ` G ) )
3		eqid 2622	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
4	3	uhgredgiedgb 26021	. . . . . 6 |- ( G e. UHGraph -> ( { A , B } e. (Edg ` G ) <-> E. i e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` i ) ) )
5	2, 4	syl5bb 272	. . . . 5 |- ( G e. UHGraph -> ( { A , B } e. E <-> E. i e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` i ) ) )
6	1	eleq2i 2693	. . . . . 6 |- ( { B , C } e. E <-> { B , C } e. (Edg ` G ) )
7	3	uhgredgiedgb 26021	. . . . . 6 |- ( G e. UHGraph -> ( { B , C } e. (Edg ` G ) <-> E. j e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` j ) ) )
8	6, 7	syl5bb 272	. . . . 5 |- ( G e. UHGraph -> ( { B , C } e. E <-> E. j e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` j ) ) )
9	1	eleq2i 2693	. . . . . 6 |- ( { C , A } e. E <-> { C , A } e. (Edg ` G ) )
10	3	uhgredgiedgb 26021	. . . . . 6 |- ( G e. UHGraph -> ( { C , A } e. (Edg ` G ) <-> E. k e. dom (iEdg ` G ) { C , A } = ( (iEdg ` G ) ` k ) ) )
11	9, 10	syl5bb 272	. . . . 5 |- ( G e. UHGraph -> ( { C , A } e. E <-> E. k e. dom (iEdg ` G ) { C , A } = ( (iEdg ` G ) ` k ) ) )
12	5, 8, 11	3anbi123d 1399	. . . 4 |- ( G e. UHGraph -> ( ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) <-> ( E. i e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` i ) /\ E. j e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` j ) /\ E. k e. dom (iEdg ` G ) { C , A } = ( (iEdg ` G ) ` k ) ) ) )
13	12	adantr 481	. . 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 /\ { C , A } e. E ) <-> ( E. i e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` i ) /\ E. j e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` j ) /\ E. k e. dom (iEdg ` G ) { C , A } = ( (iEdg ` G ) ` k ) ) ) )
14		eqid 2622	. . . . . . . . . . . . . 14 |- <" A B C A "> = <" A B C A ">
15		eqid 2622	. . . . . . . . . . . . . 14 |- <" i j k "> = <" i j k ">
16		3simpa 1058	. . . . . . . . . . . . . . . . 17 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( A e. V /\ B e. V ) )
17		pm3.22 465	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. V /\ C e. V ) -> ( C e. V /\ A e. V ) )
18	17	3adant2 1080	. . . . . . . . . . . . . . . . 17 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( C e. V /\ A e. V ) )
19	16, 18	jca 554	. . . . . . . . . . . . . . . 16 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ A e. V ) ) )
20	19	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ A e. V ) ) )
21	20	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) /\ ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ A e. V ) ) )
22		3simpa 1058	. . . . . . . . . . . . . . . . 17 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( A =/= B /\ A =/= C ) )
23		necom 2847	. . . . . . . . . . . . . . . . . . . . 21 |- ( A =/= B <-> B =/= A )
24	23	biimpi 206	. . . . . . . . . . . . . . . . . . . 20 |- ( A =/= B -> B =/= A )
25	24	anim1i 592	. . . . . . . . . . . . . . . . . . 19 |- ( ( A =/= B /\ B =/= C ) -> ( B =/= A /\ B =/= C ) )
26	25	ancomd 467	. . . . . . . . . . . . . . . . . 18 |- ( ( A =/= B /\ B =/= C ) -> ( B =/= C /\ B =/= A ) )
27	26	3adant2 1080	. . . . . . . . . . . . . . . . 17 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( B =/= C /\ B =/= A ) )
28		necom 2847	. . . . . . . . . . . . . . . . . . 19 |- ( A =/= C <-> C =/= A )
29	28	biimpi 206	. . . . . . . . . . . . . . . . . 18 |- ( A =/= C -> C =/= A )
30	29	3ad2ant2 1083	. . . . . . . . . . . . . . . . 17 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> C =/= A )
31	22, 27, 30	3jca 1242	. . . . . . . . . . . . . . . 16 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= A ) /\ C =/= A ) )
32	31	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= A ) /\ C =/= A ) )
33	32	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) /\ ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= A ) /\ C =/= A ) )
34		eqimss 3657	. . . . . . . . . . . . . . . . . 18 |- ( { A , B } = ( (iEdg ` G ) ` i ) -> { A , B } C_ ( (iEdg ` G ) ` i ) )
35	34	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) -> { A , B } C_ ( (iEdg ` G ) ` i ) )
36	35	3ad2ant3 1084	. . . . . . . . . . . . . . . 16 |- ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) -> { A , B } C_ ( (iEdg ` G ) ` i ) )
37		eqimss 3657	. . . . . . . . . . . . . . . . . 18 |- ( { B , C } = ( (iEdg ` G ) ` j ) -> { B , C } C_ ( (iEdg ` G ) ` j ) )
38	37	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) -> { B , C } C_ ( (iEdg ` G ) ` j ) )
39	38	3ad2ant1 1082	. . . . . . . . . . . . . . . 16 |- ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) -> { B , C } C_ ( (iEdg ` G ) ` j ) )
40		eqimss 3657	. . . . . . . . . . . . . . . . . 18 |- ( { C , A } = ( (iEdg ` G ) ` k ) -> { C , A } C_ ( (iEdg ` G ) ` k ) )
41	40	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) -> { C , A } C_ ( (iEdg ` G ) ` k ) )
42	41	3ad2ant2 1083	. . . . . . . . . . . . . . . 16 |- ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) -> { C , A } C_ ( (iEdg ` G ) ` k ) )
43	36, 39, 42	3jca 1242	. . . . . . . . . . . . . . 15 |- ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) -> ( { A , B } C_ ( (iEdg ` G ) ` i ) /\ { B , C } C_ ( (iEdg ` G ) ` j ) /\ { C , A } C_ ( (iEdg ` G ) ` k ) ) )
44	43	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) /\ ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> ( { A , B } C_ ( (iEdg ` G ) ` i ) /\ { B , C } C_ ( (iEdg ` G ) ` j ) /\ { C , A } C_ ( (iEdg ` G ) ` k ) ) )
45		uhgr3cyclex.v	. . . . . . . . . . . . . 14 |- V = (Vtx ` G )
46		simp3 1063	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. V /\ B e. V /\ C e. V ) -> C e. V )
47		simp1 1061	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. V /\ B e. V /\ C e. V ) -> A e. V )
48	46, 47	jca 554	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( C e. V /\ A e. V ) )
49	48, 30	anim12i 590	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( C e. V /\ A e. V ) /\ C =/= A ) )
50	49	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> ( ( C e. V /\ A e. V ) /\ C =/= A ) )
51		pm3.22 465	. . . . . . . . . . . . . . . . 17 |- ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) -> ( ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) /\ ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) ) )
52	51	3adant2 1080	. . . . . . . . . . . . . . . 16 |- ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) -> ( ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) /\ ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) ) )
53	45, 1, 3	uhgr3cyclexlem 27041	. . . . . . . . . . . . . . . 16 |- ( ( ( ( C e. V /\ A e. V ) /\ C =/= A ) /\ ( ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) /\ ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) ) ) -> i =/= j )
54	50, 52, 53	syl2an 494	. . . . . . . . . . . . . . 15 |- ( ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) /\ ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> i =/= j )
55		3simpc 1060	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( B e. V /\ C e. V ) )
56		simp3 1063	. . . . . . . . . . . . . . . . . 18 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> B =/= C )
57	55, 56	anim12i 590	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( B e. V /\ C e. V ) /\ B =/= C ) )
58	57	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> ( ( B e. V /\ C e. V ) /\ B =/= C ) )
59		3simpc 1060	. . . . . . . . . . . . . . . 16 |- ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) -> ( ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) )
60	45, 1, 3	uhgr3cyclexlem 27041	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( B e. V /\ C e. V ) /\ B =/= C ) /\ ( ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> k =/= i )
61	60	necomd 2849	. . . . . . . . . . . . . . . 16 |- ( ( ( ( B e. V /\ C e. V ) /\ B =/= C ) /\ ( ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> i =/= k )
62	58, 59, 61	syl2an 494	. . . . . . . . . . . . . . 15 |- ( ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) /\ ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> i =/= k )
63	45, 1, 3	uhgr3cyclexlem 27041	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) ) ) -> j =/= k )
64	63	exp31 630	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( A e. V /\ B e. V ) -> ( A =/= B -> ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) ) -> j =/= k ) ) )
65	64	3adant3 1081	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( A =/= B -> ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) ) -> j =/= k ) ) )
66	65	com12 32	. . . . . . . . . . . . . . . . . . . . 21 |- ( A =/= B -> ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) ) -> j =/= k ) ) )
67	66	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) ) -> j =/= k ) ) )
68	67	impcom 446	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) ) -> j =/= k ) )
69	68	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) ) -> j =/= k ) )
70	69	com12 32	. . . . . . . . . . . . . . . . 17 |- ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) ) -> ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> j =/= k ) )
71	70	3adant3 1081	. . . . . . . . . . . . . . . 16 |- ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) -> ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> j =/= k ) )
72	71	impcom 446	. . . . . . . . . . . . . . 15 |- ( ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) /\ ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> j =/= k )
73	54, 62, 72	3jca 1242	. . . . . . . . . . . . . 14 |- ( ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) /\ ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> ( i =/= j /\ i =/= k /\ j =/= k ) )
74		eqidd 2623	. . . . . . . . . . . . . 14 |- ( ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) /\ ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> A = A )
75	14, 15, 21, 33, 44, 45, 3, 73, 74	3cyclpd 27039	. . . . . . . . . . . . 13 |- ( ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) /\ ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> ( <" i j k "> (Cycles ` G ) <" A B C A "> /\ ( # ` <" i j k "> ) = 3 /\ ( <" A B C A "> ` 0 ) = A ) )
76		s3cli 13626	. . . . . . . . . . . . . . 15 |- <" i j k "> e. Word _V
77	76	elexi 3213	. . . . . . . . . . . . . 14 |- <" i j k "> e. _V
78		s4cli 13627	. . . . . . . . . . . . . . 15 |- <" A B C A "> e. Word _V
79	78	elexi 3213	. . . . . . . . . . . . . 14 |- <" A B C A "> e. _V
80		breq12 4658	. . . . . . . . . . . . . . 15 |- ( ( f = <" i j k "> /\ p = <" A B C A "> ) -> ( f (Cycles ` G ) p <-> <" i j k "> (Cycles ` G ) <" A B C A "> ) )
81		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( f = <" i j k "> -> ( # ` f ) = ( # ` <" i j k "> ) )
82	81	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( f = <" i j k "> -> ( ( # ` f ) = 3 <-> ( # ` <" i j k "> ) = 3 ) )
83	82	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( f = <" i j k "> /\ p = <" A B C A "> ) -> ( ( # ` f ) = 3 <-> ( # ` <" i j k "> ) = 3 ) )
84		fveq1 6190	. . . . . . . . . . . . . . . . 17 |- ( p = <" A B C A "> -> ( p ` 0 ) = ( <" A B C A "> ` 0 ) )
85	84	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( p = <" A B C A "> -> ( ( p ` 0 ) = A <-> ( <" A B C A "> ` 0 ) = A ) )
86	85	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( f = <" i j k "> /\ p = <" A B C A "> ) -> ( ( p ` 0 ) = A <-> ( <" A B C A "> ` 0 ) = A ) )
87	80, 83, 86	3anbi123d 1399	. . . . . . . . . . . . . 14 |- ( ( f = <" i j k "> /\ p = <" A B C A "> ) -> ( ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) <-> ( <" i j k "> (Cycles ` G ) <" A B C A "> /\ ( # ` <" i j k "> ) = 3 /\ ( <" A B C A "> ` 0 ) = A ) ) )
88	77, 79, 87	spc2ev 3301	. . . . . . . . . . . . 13 |- ( ( <" i j k "> (Cycles ` G ) <" A B C A "> /\ ( # ` <" i j k "> ) = 3 /\ ( <" A B C A "> ` 0 ) = A ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) )
89	75, 88	syl 17	. . . . . . . . . . . 12 |- ( ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) /\ ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) )
90	89	expcom 451	. . . . . . . . . . 11 |- ( ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) /\ ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) /\ ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) ) -> ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) ) )
91	90	3exp 1264	. . . . . . . . . 10 |- ( ( j e. dom (iEdg ` G ) /\ { B , C } = ( (iEdg ` G ) ` j ) ) -> ( ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) -> ( ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) -> ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) ) ) ) )
92	91	rexlimiva 3028	. . . . . . . . 9 |- ( E. j e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` j ) -> ( ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) -> ( ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) -> ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) ) ) ) )
93	92	com12 32	. . . . . . . 8 |- ( ( k e. dom (iEdg ` G ) /\ { C , A } = ( (iEdg ` G ) ` k ) ) -> ( E. j e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` j ) -> ( ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) -> ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) ) ) ) )
94	93	rexlimiva 3028	. . . . . . 7 |- ( E. k e. dom (iEdg ` G ) { C , A } = ( (iEdg ` G ) ` k ) -> ( E. j e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` j ) -> ( ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) -> ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) ) ) ) )
95	94	com13 88	. . . . . 6 |- ( ( i e. dom (iEdg ` G ) /\ { A , B } = ( (iEdg ` G ) ` i ) ) -> ( E. j e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` j ) -> ( E. k e. dom (iEdg ` G ) { C , A } = ( (iEdg ` G ) ` k ) -> ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) ) ) ) )
96	95	rexlimiva 3028	. . . . 5 |- ( E. i e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` i ) -> ( E. j e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` j ) -> ( E. k e. dom (iEdg ` G ) { C , A } = ( (iEdg ` G ) ` k ) -> ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) ) ) ) )
97	96	3imp 1256	. . . 4 |- ( ( E. i e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` i ) /\ E. j e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` j ) /\ E. k e. dom (iEdg ` G ) { C , A } = ( (iEdg ` G ) ` k ) ) -> ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) ) )
98	97	com12 32	. . 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 ) { A , B } = ( (iEdg ` G ) ` i ) /\ E. j e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` j ) /\ E. k e. dom (iEdg ` G ) { C , A } = ( (iEdg ` G ) ` k ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) ) )
99	13, 98	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 /\ { C , A } e. E ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) ) )
100	99	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 /\ { C , A } e. E ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {cpr 4179   class class class wbr 4653   dom cdm 5114   ` cfv 5888   0cc0 9936   3c3 11071   #chash 13117  Word cword 13291   <"cs3 13587   <"cs4 13588  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951  Cyclesccycls 26680

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-4 11081  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-s4 13595  df-edg 25940  df-uhgr 25953  df-wlks 26495  df-trls 26589  df-pths 26612  df-cycls 26682

This theorem is referenced by:  umgr3cyclex  27043