Theorem ushgredgedgloop 16210

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 2232	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
2		ushgredgedgloop.i	. . . . 5 |- I = (iEdg ` G )
3	1, 2	ushgrfm 16056	. . . 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 3322	. . 3 |- { i e. dom I | ( I ` i ) = { N } } C_ dom I
6		f1ores 5628	. . 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 2233	. . . . . 6 |- ( ( ( G e. USHGraph /\ N e. V ) /\ x e. A ) -> ( I ` x ) = ( I ` x ) )
12	10, 11	mpteq12dva 4190	. . . . 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 2277	. . . 4 |- ( ( G e. USHGraph /\ N e. V ) -> F = ( x e. { i e. dom I | ( I ` i ) = { N } } |-> ( I ` x ) ) )
14		f1f 5572	. . . . . . 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 5730	. . . . 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 2268	. . 3 |- ( ( G e. USHGraph /\ N e. V ) -> F = ( I |` { i e. dom I | ( I ` i ) = { N } } ) )
20		ushgruhgr 16062	. . . . . . . 8 |- ( G e. USHGraph -> G e. UHGraph )
21		eqid 2232	. . . . . . . . 9 |- (iEdg ` G ) = (iEdg ` G )
22	21	uhgrfun 16059	. . . . . . . 8 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
23	20, 22	syl 14	. . . . . . 7 |- ( G e. USHGraph -> Fun (iEdg ` G ) )
24	2	funeqi 5372	. . . . . . 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 5724	. . . . 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 5678	. . . . . . . . . 10 |- ( i = j -> ( ( I ` i ) = { N } <-> ( I ` j ) = { N } ) )
30	29	elrab 2972	. . . . . . . . 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 5807	. . . . . . . . . . . . . . . 16 |- ( ( Fun I /\ j e. dom I ) -> ( I ` j ) e. ran I )
33	2	eqcomi 2236	. . . . . . . . . . . . . . . . 17 |- (iEdg ` G ) = I
34	33	rneqi 4984	. . . . . . . . . . . . . . . 16 |- ran (iEdg ` G ) = ran I
35	32, 34	eleqtrrdi 2326	. . . . . . . . . . . . . . 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 1044	. . . . . . . . . . . . 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 2295	. . . . . . . . . . . . . . 15 |- ( f = ( I ` j ) -> ( f e. ran (iEdg ` G ) <-> ( I ` j ) e. ran (iEdg ` G ) ) )
39	38	eqcoms 2235	. . . . . . . . . . . . . 14 |- ( ( I ` j ) = f -> ( f e. ran (iEdg ` G ) <-> ( I ` j ) e. ran (iEdg ` G ) ) )
40	39	3ad2ant3 1047	. . . . . . . . . . . . 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 16041	. . . . . . . . . . . . . . . 16 |- ( G e. USHGraph -> (Edg ` G ) = ran (iEdg ` G ) )
44	42, 43	eqtrid 2277	. . . . . . . . . . . . . . 15 |- ( G e. USHGraph -> E = ran (iEdg ` G ) )
45	44	eleq2d 2302	. . . . . . . . . . . . . 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 1045	. . . . . . . . . . . 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 2239	. . . . . . . . . . . . . . 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 1220	. . . . . . . . . . 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 1229	. . . . . . . . 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 2659	. . . . . . 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 5382	. . . . . . . . . . . 12 |- ( G e. USHGraph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
59		fvelrnb 5723	. . . . . . . . . . . 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 4956	. . . . . . . . . . . . . . . . . . . . . 22 |- dom (iEdg ` G ) = dom I
62	61	eleq2i 2299	. . . . . . . . . . . . . . . . . . . . 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 5670	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( (iEdg ` G ) ` j ) = ( I ` j )
67	66	eqeq2i 2243	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f = ( (iEdg ` G ) ` j ) <-> f = ( I ` j ) )
68	67	biimpi 120	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f = ( (iEdg ` G ) ` j ) -> f = ( I ` j ) )
69	68	eqcoms 2235	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( (iEdg ` G ) ` j ) = f -> f = ( I ` j ) )
70	69	eqeq1d 2241	. . . . . . . . . . . . . . . . . . . . . 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 2240	. . . . . . . . . . . . . . . . . . 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 2641	. . . . . . . . . . . . 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 2815	. . . . . . 7 |- f e. _V
92		eqeq2 2242	. . . . . . . 8 |- ( e = f -> ( ( I ` j ) = e <-> ( I ` j ) = f ) )
93	92	rexbidv 2543	. . . . . . 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 2960	. . . . . 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 2239	. . . . . . 7 |- ( e = f -> ( e = { N } <-> f = { N } ) )
96		ushgredgedgloop.b	. . . . . . 7 |- B = { e e. E | e = { N } }
97	95, 96	elrab2 2975	. . . . . 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 2230	. . . 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 2266	. . 3 |- ( ( G e. USHGraph /\ N e. V ) -> B = ( I " { i e. dom I | ( I ` i ) = { N } } ) )
101	19, 10, 100	f1oeq123d 5607	. 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 1005   = wceq 1398   E.wex 1541   e. wcel 2203   {cab 2218   E.wrex 2521   {crab 2524   C_ wss 3210   ~Pcpw 3668   {csn 3688   |-> cmpt 4170   dom cdm 4748   ran crn 4749   |` cres 4750   "cima 4751   Fun wfun 5345   Fn wfn 5346   -->wf 5347   -1-1->wf1 5348   -1-1-onto->wf1o 5350   ` cfv 5351  Vtxcvtx 15994  iEdgciedg 15995  Edgcedg 16039  UHGraphcuhgr 16049  USHGraphcushgr 16050

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-sub 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-edg 16040  df-uhgrm 16051  df-ushgrm 16052

This theorem is referenced by:  vtxduspgrfvedgfi  16283