Theorem ushgredgedgloop 16082

Description: In a simple hypergraph there is a 1-1 onto mapping between the indexed edges being loops at a fixed vertex N and the set of loops at this vertex N. (Contributed by AV, 11-Dec-2020.) (Revised by AV, 6-Jul-2022.)

Hypotheses

Ref	Expression
ushgredgedgloop.e	|- E = (Edg ` G )
ushgredgedgloop.i	|- I = (iEdg ` G )
ushgredgedgloop.a	|- A = { i e. dom I | ( I ` i ) = { N } }
ushgredgedgloop.b	|- B = { e e. E | e = { N } }
ushgredgedgloop.f	|- F = ( x e. A |-> ( I ` x ) )

Assertion

Ref	Expression
ushgredgedgloop	|- ( ( G e. USHGraph /\ N e. V ) -> F : A -1-1-onto-> B )

Distinct variable groups:   B, e   e, E, i   e, G, i, x   e, I, i, x   e, N, i, x   e, V, i, x

Allowed substitution hints:   A( x, e, i)   B( x, i)   E( x)   F( x, e, i)

Proof of Theorem ushgredgedgloop

Dummy variables f j p w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2231	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
2		ushgredgedgloop.i	. . . . 5 |- I = (iEdg ` G )
3	1, 2	ushgrfm 15928	. . . 4 |- ( G e. USHGraph -> I : dom I -1-1-> { p e. ~P (Vtx ` G ) | E. w w e. p } )
4	3	adantr 276	. . 3 |- ( ( G e. USHGraph /\ N e. V ) -> I : dom I -1-1-> { p e. ~P (Vtx ` G ) | E. w w e. p } )
5		ssrab2 3312	. . 3 |- { i e. dom I | ( I ` i ) = { N } } C_ dom I
6		f1ores 5598	. . 3 |- ( ( I : dom I -1-1-> { p e. ~P (Vtx ` G ) | E. w w e. p } /\ { i e. dom I | ( I ` i ) = { N } } C_ dom I ) -> ( I |` { i e. dom I | ( I ` i ) = { N } } ) : { i e. dom I | ( I ` i ) = { N } } -1-1-onto-> ( I " { i e. dom I | ( I ` i ) = { N } } ) )
7	4, 5, 6	sylancl 413	. 2 |- ( ( G e. USHGraph /\ N e. V ) -> ( I |` { i e. dom I | ( I ` i ) = { N } } ) : { i e. dom I | ( I ` i ) = { N } } -1-1-onto-> ( I " { i e. dom I | ( I ` i ) = { N } } ) )
8		ushgredgedgloop.f	. . . . 5 |- F = ( x e. A |-> ( I ` x ) )
9		ushgredgedgloop.a	. . . . . . 7 |- A = { i e. dom I | ( I ` i ) = { N } }
10	9	a1i 9	. . . . . 6 |- ( ( G e. USHGraph /\ N e. V ) -> A = { i e. dom I | ( I ` i ) = { N } } )
11		eqidd 2232	. . . . . 6 |- ( ( ( G e. USHGraph /\ N e. V ) /\ x e. A ) -> ( I ` x ) = ( I ` x ) )
12	10, 11	mpteq12dva 4170	. . . . 5 |- ( ( G e. USHGraph /\ N e. V ) -> ( x e. A |-> ( I ` x ) ) = ( x e. { i e. dom I | ( I ` i ) = { N } } |-> ( I ` x ) ) )
13	8, 12	eqtrid 2276	. . . 4 |- ( ( G e. USHGraph /\ N e. V ) -> F = ( x e. { i e. dom I | ( I ` i ) = { N } } |-> ( I ` x ) ) )
14		f1f 5542	. . . . . . 7 |- ( I : dom I -1-1-> { p e. ~P (Vtx ` G ) | E. w w e. p } -> I : dom I --> { p e. ~P (Vtx ` G ) | E. w w e. p } )
15	3, 14	syl 14	. . . . . 6 |- ( G e. USHGraph -> I : dom I --> { p e. ~P (Vtx ` G ) | E. w w e. p } )
16	5	a1i 9	. . . . . 6 |- ( G e. USHGraph -> { i e. dom I | ( I ` i ) = { N } } C_ dom I )
17	15, 16	feqresmpt 5700	. . . . 5 |- ( G e. USHGraph -> ( I |` { i e. dom I | ( I ` i ) = { N } } ) = ( x e. { i e. dom I | ( I ` i ) = { N } } |-> ( I ` x ) ) )
18	17	adantr 276	. . . 4 |- ( ( G e. USHGraph /\ N e. V ) -> ( I |` { i e. dom I | ( I ` i ) = { N } } ) = ( x e. { i e. dom I | ( I ` i ) = { N } } |-> ( I ` x ) ) )
19	13, 18	eqtr4d 2267	. . 3 |- ( ( G e. USHGraph /\ N e. V ) -> F = ( I |` { i e. dom I | ( I ` i ) = { N } } ) )
20		ushgruhgr 15934	. . . . . . . 8 |- ( G e. USHGraph -> G e. UHGraph )
21		eqid 2231	. . . . . . . . 9 |- (iEdg ` G ) = (iEdg ` G )
22	21	uhgrfun 15931	. . . . . . . 8 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
23	20, 22	syl 14	. . . . . . 7 |- ( G e. USHGraph -> Fun (iEdg ` G ) )
24	2	funeqi 5347	. . . . . . 7 |- ( Fun I <-> Fun (iEdg ` G ) )
25	23, 24	sylibr 134	. . . . . 6 |- ( G e. USHGraph -> Fun I )
26	25	adantr 276	. . . . 5 |- ( ( G e. USHGraph /\ N e. V ) -> Fun I )
27		dfimafn 5694	. . . . 5 |- ( ( Fun I /\ { i e. dom I | ( I ` i ) = { N } } C_ dom I ) -> ( I " { i e. dom I | ( I ` i ) = { N } } ) = { e | E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e } )
28	26, 5, 27	sylancl 413	. . . 4 |- ( ( G e. USHGraph /\ N e. V ) -> ( I " { i e. dom I | ( I ` i ) = { N } } ) = { e | E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e } )
29		fveqeq2 5648	. . . . . . . . . 10 |- ( i = j -> ( ( I ` i ) = { N } <-> ( I ` j ) = { N } ) )
30	29	elrab 2962	. . . . . . . . 9 |- ( j e. { i e. dom I | ( I ` i ) = { N } } <-> ( j e. dom I /\ ( I ` j ) = { N } ) )
31		simpl 109	. . . . . . . . . . . . . . 15 |- ( ( j e. dom I /\ ( I ` j ) = { N } ) -> j e. dom I )
32		fvelrn 5778	. . . . . . . . . . . . . . . 16 |- ( ( Fun I /\ j e. dom I ) -> ( I ` j ) e. ran I )
33	2	eqcomi 2235	. . . . . . . . . . . . . . . . 17 |- (iEdg ` G ) = I
34	33	rneqi 4960	. . . . . . . . . . . . . . . 16 |- ran (iEdg ` G ) = ran I
35	32, 34	eleqtrrdi 2325	. . . . . . . . . . . . . . 15 |- ( ( Fun I /\ j e. dom I ) -> ( I ` j ) e. ran (iEdg ` G ) )
36	26, 31, 35	syl2an 289	. . . . . . . . . . . . . 14 |- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) ) -> ( I ` j ) e. ran (iEdg ` G ) )
37	36	3adant3 1043	. . . . . . . . . . . . 13 |- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> ( I ` j ) e. ran (iEdg ` G ) )
38		eleq1 2294	. . . . . . . . . . . . . . 15 |- ( f = ( I ` j ) -> ( f e. ran (iEdg ` G ) <-> ( I ` j ) e. ran (iEdg ` G ) ) )
39	38	eqcoms 2234	. . . . . . . . . . . . . 14 |- ( ( I ` j ) = f -> ( f e. ran (iEdg ` G ) <-> ( I ` j ) e. ran (iEdg ` G ) ) )
40	39	3ad2ant3 1046	. . . . . . . . . . . . 13 |- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> ( f e. ran (iEdg ` G ) <-> ( I ` j ) e. ran (iEdg ` G ) ) )
41	37, 40	mpbird 167	. . . . . . . . . . . 12 |- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> f e. ran (iEdg ` G ) )
42		ushgredgedgloop.e	. . . . . . . . . . . . . . . 16 |- E = (Edg ` G )
43		edgvalg 15913	. . . . . . . . . . . . . . . 16 |- ( G e. USHGraph -> (Edg ` G ) = ran (iEdg ` G ) )
44	42, 43	eqtrid 2276	. . . . . . . . . . . . . . 15 |- ( G e. USHGraph -> E = ran (iEdg ` G ) )
45	44	eleq2d 2301	. . . . . . . . . . . . . 14 |- ( G e. USHGraph -> ( f e. E <-> f e. ran (iEdg ` G ) ) )
46	45	adantr 276	. . . . . . . . . . . . 13 |- ( ( G e. USHGraph /\ N e. V ) -> ( f e. E <-> f e. ran (iEdg ` G ) ) )
47	46	3ad2ant1 1044	. . . . . . . . . . . 12 |- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> ( f e. E <-> f e. ran (iEdg ` G ) ) )
48	41, 47	mpbird 167	. . . . . . . . . . 11 |- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> f e. E )
49		eqeq1 2238	. . . . . . . . . . . . . . 15 |- ( ( I ` j ) = f -> ( ( I ` j ) = { N } <-> f = { N } ) )
50	49	biimpcd 159	. . . . . . . . . . . . . 14 |- ( ( I ` j ) = { N } -> ( ( I ` j ) = f -> f = { N } ) )
51	50	adantl 277	. . . . . . . . . . . . 13 |- ( ( j e. dom I /\ ( I ` j ) = { N } ) -> ( ( I ` j ) = f -> f = { N } ) )
52	51	a1i 9	. . . . . . . . . . . 12 |- ( ( G e. USHGraph /\ N e. V ) -> ( ( j e. dom I /\ ( I ` j ) = { N } ) -> ( ( I ` j ) = f -> f = { N } ) ) )
53	52	3imp 1219	. . . . . . . . . . 11 |- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> f = { N } )
54	48, 53	jca 306	. . . . . . . . . 10 |- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> ( f e. E /\ f = { N } ) )
55	54	3exp 1228	. . . . . . . . 9 |- ( ( G e. USHGraph /\ N e. V ) -> ( ( j e. dom I /\ ( I ` j ) = { N } ) -> ( ( I ` j ) = f -> ( f e. E /\ f = { N } ) ) ) )
56	30, 55	biimtrid 152	. . . . . . . 8 |- ( ( G e. USHGraph /\ N e. V ) -> ( j e. { i e. dom I | ( I ` i ) = { N } } -> ( ( I ` j ) = f -> ( f e. E /\ f = { N } ) ) ) )
57	56	rexlimdv 2649	. . . . . . 7 |- ( ( G e. USHGraph /\ N e. V ) -> ( E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f -> ( f e. E /\ f = { N } ) ) )
58	23	funfnd 5357	. . . . . . . . . . . 12 |- ( G e. USHGraph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
59		fvelrnb 5693	. . . . . . . . . . . 12 |- ( (iEdg ` G ) Fn dom (iEdg ` G ) -> ( f e. ran (iEdg ` G ) <-> E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = f ) )
60	58, 59	syl 14	. . . . . . . . . . 11 |- ( G e. USHGraph -> ( f e. ran (iEdg ` G ) <-> E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = f ) )
61	33	dmeqi 4932	. . . . . . . . . . . . . . . . . . . . . 22 |- dom (iEdg ` G ) = dom I
62	61	eleq2i 2298	. . . . . . . . . . . . . . . . . . . . 21 |- ( j e. dom (iEdg ` G ) <-> j e. dom I )
63	62	biimpi 120	. . . . . . . . . . . . . . . . . . . 20 |- ( j e. dom (iEdg ` G ) -> j e. dom I )
64	63	adantr 276	. . . . . . . . . . . . . . . . . . 19 |- ( ( j e. dom (iEdg ` G ) /\ ( (iEdg ` G ) ` j ) = f ) -> j e. dom I )
65	64	adantl 277	. . . . . . . . . . . . . . . . . 18 |- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom (iEdg ` G ) /\ ( (iEdg ` G ) ` j ) = f ) ) -> j e. dom I )
66	33	fveq1i 5640	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( (iEdg ` G ) ` j ) = ( I ` j )
67	66	eqeq2i 2242	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f = ( (iEdg ` G ) ` j ) <-> f = ( I ` j ) )
68	67	biimpi 120	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f = ( (iEdg ` G ) ` j ) -> f = ( I ` j ) )
69	68	eqcoms 2234	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( (iEdg ` G ) ` j ) = f -> f = ( I ` j ) )
70	69	eqeq1d 2240	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( (iEdg ` G ) ` j ) = f -> ( f = { N } <-> ( I ` j ) = { N } ) )
71	70	biimpcd 159	. . . . . . . . . . . . . . . . . . . . 21 |- ( f = { N } -> ( ( (iEdg ` G ) ` j ) = f -> ( I ` j ) = { N } ) )
72	71	adantl 277	. . . . . . . . . . . . . . . . . . . 20 |- ( ( G e. USHGraph /\ f = { N } ) -> ( ( (iEdg ` G ) ` j ) = f -> ( I ` j ) = { N } ) )
73	72	adantld 278	. . . . . . . . . . . . . . . . . . 19 |- ( ( G e. USHGraph /\ f = { N } ) -> ( ( j e. dom (iEdg ` G ) /\ ( (iEdg ` G ) ` j ) = f ) -> ( I ` j ) = { N } ) )
74	73	imp 124	. . . . . . . . . . . . . . . . . 18 |- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom (iEdg ` G ) /\ ( (iEdg ` G ) ` j ) = f ) ) -> ( I ` j ) = { N } )
75	65, 74	jca 306	. . . . . . . . . . . . . . . . 17 |- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom (iEdg ` G ) /\ ( (iEdg ` G ) ` j ) = f ) ) -> ( j e. dom I /\ ( I ` j ) = { N } ) )
76	75, 30	sylibr 134	. . . . . . . . . . . . . . . 16 |- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom (iEdg ` G ) /\ ( (iEdg ` G ) ` j ) = f ) ) -> j e. { i e. dom I | ( I ` i ) = { N } } )
77	66	eqeq1i 2239	. . . . . . . . . . . . . . . . . . 19 |- ( ( (iEdg ` G ) ` j ) = f <-> ( I ` j ) = f )
78	77	biimpi 120	. . . . . . . . . . . . . . . . . 18 |- ( ( (iEdg ` G ) ` j ) = f -> ( I ` j ) = f )
79	78	adantl 277	. . . . . . . . . . . . . . . . 17 |- ( ( j e. dom (iEdg ` G ) /\ ( (iEdg ` G ) ` j ) = f ) -> ( I ` j ) = f )
80	79	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom (iEdg ` G ) /\ ( (iEdg ` G ) ` j ) = f ) ) -> ( I ` j ) = f )
81	76, 80	jca 306	. . . . . . . . . . . . . . 15 |- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom (iEdg ` G ) /\ ( (iEdg ` G ) ` j ) = f ) ) -> ( j e. { i e. dom I | ( I ` i ) = { N } } /\ ( I ` j ) = f ) )
82	81	ex 115	. . . . . . . . . . . . . 14 |- ( ( G e. USHGraph /\ f = { N } ) -> ( ( j e. dom (iEdg ` G ) /\ ( (iEdg ` G ) ` j ) = f ) -> ( j e. { i e. dom I | ( I ` i ) = { N } } /\ ( I ` j ) = f ) ) )
83	82	reximdv2 2631	. . . . . . . . . . . . 13 |- ( ( G e. USHGraph /\ f = { N } ) -> ( E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = f -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) )
84	83	ex 115	. . . . . . . . . . . 12 |- ( G e. USHGraph -> ( f = { N } -> ( E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = f -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) )
85	84	com23 78	. . . . . . . . . . 11 |- ( G e. USHGraph -> ( E. j e. dom (iEdg ` G ) ( (iEdg ` G ) ` j ) = f -> ( f = { N } -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) )
86	60, 85	sylbid 150	. . . . . . . . . 10 |- ( G e. USHGraph -> ( f e. ran (iEdg ` G ) -> ( f = { N } -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) )
87	45, 86	sylbid 150	. . . . . . . . 9 |- ( G e. USHGraph -> ( f e. E -> ( f = { N } -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) )
88	87	impd 254	. . . . . . . 8 |- ( G e. USHGraph -> ( ( f e. E /\ f = { N } ) -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) )
89	88	adantr 276	. . . . . . 7 |- ( ( G e. USHGraph /\ N e. V ) -> ( ( f e. E /\ f = { N } ) -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) )
90	57, 89	impbid 129	. . . . . 6 |- ( ( G e. USHGraph /\ N e. V ) -> ( E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f <-> ( f e. E /\ f = { N } ) ) )
91		vex 2805	. . . . . . 7 |- f e. _V
92		eqeq2 2241	. . . . . . . 8 |- ( e = f -> ( ( I ` j ) = e <-> ( I ` j ) = f ) )
93	92	rexbidv 2533	. . . . . . 7 |- ( e = f -> ( E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e <-> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) )
94	91, 93	elab 2950	. . . . . 6 |- ( f e. { e | E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e } <-> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f )
95		eqeq1 2238	. . . . . . 7 |- ( e = f -> ( e = { N } <-> f = { N } ) )
96		ushgredgedgloop.b	. . . . . . 7 |- B = { e e. E | e = { N } }
97	95, 96	elrab2 2965	. . . . . 6 |- ( f e. B <-> ( f e. E /\ f = { N } ) )
98	90, 94, 97	3bitr4g 223	. . . . 5 |- ( ( G e. USHGraph /\ N e. V ) -> ( f e. { e | E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e } <-> f e. B ) )
99	98	eqrdv 2229	. . . 4 |- ( ( G e. USHGraph /\ N e. V ) -> { e | E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e } = B )
100	28, 99	eqtr2d 2265	. . 3 |- ( ( G e. USHGraph /\ N e. V ) -> B = ( I " { i e. dom I | ( I ` i ) = { N } } ) )
101	19, 10, 100	f1oeq123d 5577	. 2 |- ( ( G e. USHGraph /\ N e. V ) -> ( F : A -1-1-onto-> B <-> ( I |` { i e. dom I | ( I ` i ) = { N } } ) : { i e. dom I | ( I ` i ) = { N } } -1-1-onto-> ( I " { i e. dom I | ( I ` i ) = { N } } ) ) )
102	7, 101	mpbird 167	1 |- ( ( G e. USHGraph /\ N e. V ) -> F : A -1-1-onto-> B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   E.wrex 2511   {crab 2514   C_ wss 3200   ~Pcpw 3652   {csn 3669   |-> cmpt 4150   dom cdm 4725   ran crn 4726   |` cres 4727   "cima 4728   Fun wfun 5320   Fn wfn 5321   -->wf 5322   -1-1->wf1 5323   -1-1-onto->wf1o 5325   ` cfv 5326  Vtxcvtx 15866  iEdgciedg 15867  Edgcedg 15911  UHGraphcuhgr 15921  USHGraphcushgr 15922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-edg 15912  df-uhgrm 15923  df-ushgrm 15924

This theorem is referenced by:  vtxduspgrfvedgfi  16155