Theorem ushgredgedgloop 15991

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 2209	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
2		ushgredgedgloop.i	. . . . 5 |- I = (iEdg ` G )
3	1, 2	ushgrfm 15839	. . . 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 3289	. . 3 |- { i e. dom I | ( I ` i ) = { N } } C_ dom I
6		f1ores 5563	. . 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 2210	. . . . . 6 |- ( ( ( G e. USHGraph /\ N e. V ) /\ x e. A ) -> ( I ` x ) = ( I ` x ) )
12	10, 11	mpteq12dva 4144	. . . . 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 2254	. . . 4 |- ( ( G e. USHGraph /\ N e. V ) -> F = ( x e. { i e. dom I | ( I ` i ) = { N } } |-> ( I ` x ) ) )
14		f1f 5507	. . . . . . 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 5661	. . . . 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 2245	. . 3 |- ( ( G e. USHGraph /\ N e. V ) -> F = ( I |` { i e. dom I | ( I ` i ) = { N } } ) )
20		ushgruhgr 15845	. . . . . . . 8 |- ( G e. USHGraph -> G e. UHGraph )
21		eqid 2209	. . . . . . . . 9 |- (iEdg ` G ) = (iEdg ` G )
22	21	uhgrfun 15842	. . . . . . . 8 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
23	20, 22	syl 14	. . . . . . 7 |- ( G e. USHGraph -> Fun (iEdg ` G ) )
24	2	funeqi 5315	. . . . . . 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 5655	. . . . 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 5612	. . . . . . . . . 10 |- ( i = j -> ( ( I ` i ) = { N } <-> ( I ` j ) = { N } ) )
30	29	elrab 2939	. . . . . . . . 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 5739	. . . . . . . . . . . . . . . 16 |- ( ( Fun I /\ j e. dom I ) -> ( I ` j ) e. ran I )
33	2	eqcomi 2213	. . . . . . . . . . . . . . . . 17 |- (iEdg ` G ) = I
34	33	rneqi 4928	. . . . . . . . . . . . . . . 16 |- ran (iEdg ` G ) = ran I
35	32, 34	eleqtrrdi 2303	. . . . . . . . . . . . . . 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 1022	. . . . . . . . . . . . 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 2272	. . . . . . . . . . . . . . 15 |- ( f = ( I ` j ) -> ( f e. ran (iEdg ` G ) <-> ( I ` j ) e. ran (iEdg ` G ) ) )
39	38	eqcoms 2212	. . . . . . . . . . . . . 14 |- ( ( I ` j ) = f -> ( f e. ran (iEdg ` G ) <-> ( I ` j ) e. ran (iEdg ` G ) ) )
40	39	3ad2ant3 1025	. . . . . . . . . . . . 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 15825	. . . . . . . . . . . . . . . 16 |- ( G e. USHGraph -> (Edg ` G ) = ran (iEdg ` G ) )
44	42, 43	eqtrid 2254	. . . . . . . . . . . . . . 15 |- ( G e. USHGraph -> E = ran (iEdg ` G ) )
45	44	eleq2d 2279	. . . . . . . . . . . . . 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 1023	. . . . . . . . . . . 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 2216	. . . . . . . . . . . . . . 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 1198	. . . . . . . . . . 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 1207	. . . . . . . . 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 2627	. . . . . . 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 5325	. . . . . . . . . . . 12 |- ( G e. USHGraph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
59		fvelrnb 5654	. . . . . . . . . . . 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 4901	. . . . . . . . . . . . . . . . . . . . . 22 |- dom (iEdg ` G ) = dom I
62	61	eleq2i 2276	. . . . . . . . . . . . . . . . . . . . 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 5604	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( (iEdg ` G ) ` j ) = ( I ` j )
67	66	eqeq2i 2220	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f = ( (iEdg ` G ) ` j ) <-> f = ( I ` j ) )
68	67	biimpi 120	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f = ( (iEdg ` G ) ` j ) -> f = ( I ` j ) )
69	68	eqcoms 2212	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( (iEdg ` G ) ` j ) = f -> f = ( I ` j ) )
70	69	eqeq1d 2218	. . . . . . . . . . . . . . . . . . . . . 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 2217	. . . . . . . . . . . . . . . . . . 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 2609	. . . . . . . . . . . . 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 2782	. . . . . . 7 |- f e. _V
92		eqeq2 2219	. . . . . . . 8 |- ( e = f -> ( ( I ` j ) = e <-> ( I ` j ) = f ) )
93	92	rexbidv 2511	. . . . . . 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 2927	. . . . . 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 2216	. . . . . . 7 |- ( e = f -> ( e = { N } <-> f = { N } ) )
96		ushgredgedgloop.b	. . . . . . 7 |- B = { e e. E | e = { N } }
97	95, 96	elrab2 2942	. . . . . 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 2207	. . . 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 2243	. . 3 |- ( ( G e. USHGraph /\ N e. V ) -> B = ( I " { i e. dom I | ( I ` i ) = { N } } ) )
101	19, 10, 100	f1oeq123d 5542	. 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 983   = wceq 1375   E.wex 1518   e. wcel 2180   {cab 2195   E.wrex 2489   {crab 2492   C_ wss 3177   ~Pcpw 3629   {csn 3646   |-> cmpt 4124   dom cdm 4696   ran crn 4697   |` cres 4698   "cima 4699   Fun wfun 5288   Fn wfn 5289   -->wf 5290   -1-1->wf1 5291   -1-1-onto->wf1o 5293   ` cfv 5294  Vtxcvtx 15778  iEdgciedg 15779  Edgcedg 15823  UHGraphcuhgr 15832  USHGraphcushgr 15833

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-sub 8287  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-dec 9547  df-ndx 13001  df-slot 13002  df-base 13004  df-edgf 15771  df-vtx 15780  df-iedg 15781  df-edg 15824  df-uhgrm 15834  df-ushgrm 15835

This theorem is referenced by: (None)