Theorem rankcf 9058

Description: Any set must be at least as large as the cofinality of its rank, because the ranks of the elements of A form a cofinal map into ( rank ` A ). (Contributed by Mario Carneiro, 27-May-2013.)

Assertion

Ref	Expression
rankcf	|- -. A ~< ( cf ` ( rank ` A ) )

Proof of Theorem rankcf

Dummy variables w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rankon 8116	. . 3 |- ( rank ` A ) e. On
2		onzsl 6570	. . 3 |- ( ( rank ` A ) e. On <-> ( ( rank ` A ) = (/) \/ E. x e. On ( rank ` A ) = suc x \/ ( ( rank ` A ) e. _V /\ Lim ( rank ` A ) ) ) )
3	1, 2	mpbi 208	. 2 |- ( ( rank ` A ) = (/) \/ E. x e. On ( rank ` A ) = suc x \/ ( ( rank ` A ) e. _V /\ Lim ( rank ` A ) ) )
4		sdom0 7556	. . . 4 |- -. A ~< (/)
5		fveq2 5802	. . . . . 6 |- ( ( rank ` A ) = (/) -> ( cf ` ( rank ` A ) ) = ( cf ` (/) ) )
6		cf0 8534	. . . . . 6 |- ( cf ` (/) ) = (/)
7	5, 6	syl6eq 2511	. . . . 5 |- ( ( rank ` A ) = (/) -> ( cf ` ( rank ` A ) ) = (/) )
8	7	breq2d 4415	. . . 4 |- ( ( rank ` A ) = (/) -> ( A ~< ( cf ` ( rank ` A ) ) <-> A ~< (/) ) )
9	4, 8	mtbiri 303	. . 3 |- ( ( rank ` A ) = (/) -> -. A ~< ( cf ` ( rank ` A ) ) )
10		fveq2 5802	. . . . . . 7 |- ( ( rank ` A ) = suc x -> ( cf ` ( rank ` A ) ) = ( cf ` suc x ) )
11		cfsuc 8540	. . . . . . 7 |- ( x e. On -> ( cf ` suc x ) = 1o )
12	10, 11	sylan9eqr 2517	. . . . . 6 |- ( ( x e. On /\ ( rank ` A ) = suc x ) -> ( cf ` ( rank ` A ) ) = 1o )
13		nsuceq0 4910	. . . . . . . . 9 |- suc x =/= (/)
14		neeq1 2733	. . . . . . . . 9 |- ( ( rank ` A ) = suc x -> ( ( rank ` A ) =/= (/) <-> suc x =/= (/) ) )
15	13, 14	mpbiri 233	. . . . . . . 8 |- ( ( rank ` A ) = suc x -> ( rank ` A ) =/= (/) )
16		fveq2 5802	. . . . . . . . . . 11 |- ( A = (/) -> ( rank ` A ) = ( rank ` (/) ) )
17		0elon 4883	. . . . . . . . . . . . 13 |- (/) e. On
18		r1fnon 8088	. . . . . . . . . . . . . 14 |- R1 Fn On
19		fndm 5621	. . . . . . . . . . . . . 14 |- ( R1 Fn On -> dom R1 = On )
20	18, 19	ax-mp 5	. . . . . . . . . . . . 13 |- dom R1 = On
21	17, 20	eleqtrri 2541	. . . . . . . . . . . 12 |- (/) e. dom R1
22		rankonid 8150	. . . . . . . . . . . 12 |- ( (/) e. dom R1 <-> ( rank ` (/) ) = (/) )
23	21, 22	mpbi 208	. . . . . . . . . . 11 |- ( rank ` (/) ) = (/)
24	16, 23	syl6eq 2511	. . . . . . . . . 10 |- ( A = (/) -> ( rank ` A ) = (/) )
25	24	necon3i 2692	. . . . . . . . 9 |- ( ( rank ` A ) =/= (/) -> A =/= (/) )
26		rankvaln 8120	. . . . . . . . . . 11 |- ( -. A e. U. ( R1 " On ) -> ( rank ` A ) = (/) )
27	26	necon1ai 2683	. . . . . . . . . 10 |- ( ( rank ` A ) =/= (/) -> A e. U. ( R1 " On ) )
28		breq2 4407	. . . . . . . . . . 11 |- ( y = A -> ( 1o ~<_ y <-> 1o ~<_ A ) )
29		neeq1 2733	. . . . . . . . . . 11 |- ( y = A -> ( y =/= (/) <-> A =/= (/) ) )
30		0sdom1dom 7624	. . . . . . . . . . . 12 |- ( (/) ~< y <-> 1o ~<_ y )
31		vex 3081	. . . . . . . . . . . . 13 |- y e. _V
32	31	0sdom 7555	. . . . . . . . . . . 12 |- ( (/) ~< y <-> y =/= (/) )
33	30, 32	bitr3i 251	. . . . . . . . . . 11 |- ( 1o ~<_ y <-> y =/= (/) )
34	28, 29, 33	vtoclbg 3137	. . . . . . . . . 10 |- ( A e. U. ( R1 " On ) -> ( 1o ~<_ A <-> A =/= (/) ) )
35	27, 34	syl 16	. . . . . . . . 9 |- ( ( rank ` A ) =/= (/) -> ( 1o ~<_ A <-> A =/= (/) ) )
36	25, 35	mpbird 232	. . . . . . . 8 |- ( ( rank ` A ) =/= (/) -> 1o ~<_ A )
37	15, 36	syl 16	. . . . . . 7 |- ( ( rank ` A ) = suc x -> 1o ~<_ A )
38	37	adantl 466	. . . . . 6 |- ( ( x e. On /\ ( rank ` A ) = suc x ) -> 1o ~<_ A )
39	12, 38	eqbrtrd 4423	. . . . 5 |- ( ( x e. On /\ ( rank ` A ) = suc x ) -> ( cf ` ( rank ` A ) ) ~<_ A )
40	39	rexlimiva 2942	. . . 4 |- ( E. x e. On ( rank ` A ) = suc x -> ( cf ` ( rank ` A ) ) ~<_ A )
41		domnsym 7550	. . . 4 |- ( ( cf ` ( rank ` A ) ) ~<_ A -> -. A ~< ( cf ` ( rank ` A ) ) )
42	40, 41	syl 16	. . 3 |- ( E. x e. On ( rank ` A ) = suc x -> -. A ~< ( cf ` ( rank ` A ) ) )
43		nlim0 4888	. . . . . . . . . . . . . . . . 17 |- -. Lim (/)
44		limeq 4842	. . . . . . . . . . . . . . . . 17 |- ( ( rank ` A ) = (/) -> ( Lim ( rank ` A ) <-> Lim (/) ) )
45	43, 44	mtbiri 303	. . . . . . . . . . . . . . . 16 |- ( ( rank ` A ) = (/) -> -. Lim ( rank ` A ) )
46	26, 45	syl 16	. . . . . . . . . . . . . . 15 |- ( -. A e. U. ( R1 " On ) -> -. Lim ( rank ` A ) )
47	46	con4i 130	. . . . . . . . . . . . . 14 |- ( Lim ( rank ` A ) -> A e. U. ( R1 " On ) )
48		r1elssi 8126	. . . . . . . . . . . . . 14 |- ( A e. U. ( R1 " On ) -> A C_ U. ( R1 " On ) )
49	47, 48	syl 16	. . . . . . . . . . . . 13 |- ( Lim ( rank ` A ) -> A C_ U. ( R1 " On ) )
50	49	sselda 3467	. . . . . . . . . . . 12 |- ( ( Lim ( rank ` A ) /\ x e. A ) -> x e. U. ( R1 " On ) )
51		ranksnb 8148	. . . . . . . . . . . 12 |- ( x e. U. ( R1 " On ) -> ( rank ` { x } ) = suc ( rank ` x ) )
52	50, 51	syl 16	. . . . . . . . . . 11 |- ( ( Lim ( rank ` A ) /\ x e. A ) -> ( rank ` { x } ) = suc ( rank ` x ) )
53		rankelb 8145	. . . . . . . . . . . . . 14 |- ( A e. U. ( R1 " On ) -> ( x e. A -> ( rank ` x ) e. ( rank ` A ) ) )
54	47, 53	syl 16	. . . . . . . . . . . . 13 |- ( Lim ( rank ` A ) -> ( x e. A -> ( rank ` x ) e. ( rank ` A ) ) )
55		limsuc 6573	. . . . . . . . . . . . 13 |- ( Lim ( rank ` A ) -> ( ( rank ` x ) e. ( rank ` A ) <-> suc ( rank ` x ) e. ( rank ` A ) ) )
56	54, 55	sylibd 214	. . . . . . . . . . . 12 |- ( Lim ( rank ` A ) -> ( x e. A -> suc ( rank ` x ) e. ( rank ` A ) ) )
57	56	imp 429	. . . . . . . . . . 11 |- ( ( Lim ( rank ` A ) /\ x e. A ) -> suc ( rank ` x ) e. ( rank ` A ) )
58	52, 57	eqeltrd 2542	. . . . . . . . . 10 |- ( ( Lim ( rank ` A ) /\ x e. A ) -> ( rank ` { x } ) e. ( rank ` A ) )
59		eleq1a 2537	. . . . . . . . . 10 |- ( ( rank ` { x } ) e. ( rank ` A ) -> ( w = ( rank ` { x } ) -> w e. ( rank ` A ) ) )
60	58, 59	syl 16	. . . . . . . . 9 |- ( ( Lim ( rank ` A ) /\ x e. A ) -> ( w = ( rank ` { x } ) -> w e. ( rank ` A ) ) )
61	60	rexlimdva 2947	. . . . . . . 8 |- ( Lim ( rank ` A ) -> ( E. x e. A w = ( rank ` { x } ) -> w e. ( rank ` A ) ) )
62	61	abssdv 3537	. . . . . . 7 |- ( Lim ( rank ` A ) -> { w | E. x e. A w = ( rank ` { x } ) } C_ ( rank ` A ) )
63		snex 4644	. . . . . . . . . . . . 13 |- { x } e. _V
64	63	dfiun2 4315	. . . . . . . . . . . 12 |- U_ x e. A { x } = U. { y | E. x e. A y = { x } }
65		iunid 4336	. . . . . . . . . . . 12 |- U_ x e. A { x } = A
66	64, 65	eqtr3i 2485	. . . . . . . . . . 11 |- U. { y | E. x e. A y = { x } } = A
67	66	fveq2i 5805	. . . . . . . . . 10 |- ( rank ` U. { y | E. x e. A y = { x } } ) = ( rank ` A )
68	48	sselda 3467	. . . . . . . . . . . . . . 15 |- ( ( A e. U. ( R1 " On ) /\ x e. A ) -> x e. U. ( R1 " On ) )
69		snwf 8130	. . . . . . . . . . . . . . 15 |- ( x e. U. ( R1 " On ) -> { x } e. U. ( R1 " On ) )
70		eleq1a 2537	. . . . . . . . . . . . . . 15 |- ( { x } e. U. ( R1 " On ) -> ( y = { x } -> y e. U. ( R1 " On ) ) )
71	68, 69, 70	3syl 20	. . . . . . . . . . . . . 14 |- ( ( A e. U. ( R1 " On ) /\ x e. A ) -> ( y = { x } -> y e. U. ( R1 " On ) ) )
72	71	rexlimdva 2947	. . . . . . . . . . . . 13 |- ( A e. U. ( R1 " On ) -> ( E. x e. A y = { x } -> y e. U. ( R1 " On ) ) )
73	72	abssdv 3537	. . . . . . . . . . . 12 |- ( A e. U. ( R1 " On ) -> { y | E. x e. A y = { x } } C_ U. ( R1 " On ) )
74		abrexexg 6665	. . . . . . . . . . . . 13 |- ( A e. U. ( R1 " On ) -> { y | E. x e. A y = { x } } e. _V )
75		eleq1 2526	. . . . . . . . . . . . . 14 |- ( z = { y | E. x e. A y = { x } } -> ( z e. U. ( R1 " On ) <-> { y | E. x e. A y = { x } } e. U. ( R1 " On ) ) )
76		sseq1 3488	. . . . . . . . . . . . . 14 |- ( z = { y | E. x e. A y = { x } } -> ( z C_ U. ( R1 " On ) <-> { y | E. x e. A y = { x } } C_ U. ( R1 " On ) ) )
77		vex 3081	. . . . . . . . . . . . . . 15 |- z e. _V
78	77	r1elss 8127	. . . . . . . . . . . . . 14 |- ( z e. U. ( R1 " On ) <-> z C_ U. ( R1 " On ) )
79	75, 76, 78	vtoclbg 3137	. . . . . . . . . . . . 13 |- ( { y | E. x e. A y = { x } } e. _V -> ( { y | E. x e. A y = { x } } e. U. ( R1 " On ) <-> { y | E. x e. A y = { x } } C_ U. ( R1 " On ) ) )
80	74, 79	syl 16	. . . . . . . . . . . 12 |- ( A e. U. ( R1 " On ) -> ( { y | E. x e. A y = { x } } e. U. ( R1 " On ) <-> { y | E. x e. A y = { x } } C_ U. ( R1 " On ) ) )
81	73, 80	mpbird 232	. . . . . . . . . . 11 |- ( A e. U. ( R1 " On ) -> { y | E. x e. A y = { x } } e. U. ( R1 " On ) )
82		rankuni2b 8174	. . . . . . . . . . 11 |- ( { y | E. x e. A y = { x } } e. U. ( R1 " On ) -> ( rank ` U. { y | E. x e. A y = { x } } ) = U_ z e. { y | E. x e. A y = { x } } ( rank ` z ) )
83	81, 82	syl 16	. . . . . . . . . 10 |- ( A e. U. ( R1 " On ) -> ( rank ` U. { y | E. x e. A y = { x } } ) = U_ z e. { y | E. x e. A y = { x } } ( rank ` z ) )
84	67, 83	syl5eqr 2509	. . . . . . . . 9 |- ( A e. U. ( R1 " On ) -> ( rank ` A ) = U_ z e. { y | E. x e. A y = { x } } ( rank ` z ) )
85		fvex 5812	. . . . . . . . . . 11 |- ( rank ` z ) e. _V
86	85	dfiun2 4315	. . . . . . . . . 10 |- U_ z e. { y | E. x e. A y = { x } } ( rank ` z ) = U. { w | E. z e. { y | E. x e. A y = { x } } w = ( rank ` z ) }
87		fveq2 5802	. . . . . . . . . . . 12 |- ( z = { x } -> ( rank ` z ) = ( rank ` { x } ) )
88	63, 87	abrexco 6073	. . . . . . . . . . 11 |- { w | E. z e. { y | E. x e. A y = { x } } w = ( rank ` z ) } = { w | E. x e. A w = ( rank ` { x } ) }
89	88	unieqi 4211	. . . . . . . . . 10 |- U. { w | E. z e. { y | E. x e. A y = { x } } w = ( rank ` z ) } = U. { w | E. x e. A w = ( rank ` { x } ) }
90	86, 89	eqtri 2483	. . . . . . . . 9 |- U_ z e. { y | E. x e. A y = { x } } ( rank ` z ) = U. { w | E. x e. A w = ( rank ` { x } ) }
91	84, 90	syl6req 2512	. . . . . . . 8 |- ( A e. U. ( R1 " On ) -> U. { w | E. x e. A w = ( rank ` { x } ) } = ( rank ` A ) )
92	47, 91	syl 16	. . . . . . 7 |- ( Lim ( rank ` A ) -> U. { w | E. x e. A w = ( rank ` { x } ) } = ( rank ` A ) )
93		fvex 5812	. . . . . . . 8 |- ( rank ` A ) e. _V
94	93	cfslb 8549	. . . . . . 7 |- ( ( Lim ( rank ` A ) /\ { w | E. x e. A w = ( rank ` { x } ) } C_ ( rank ` A ) /\ U. { w | E. x e. A w = ( rank ` { x } ) } = ( rank ` A ) ) -> ( cf ` ( rank ` A ) ) ~<_ { w | E. x e. A w = ( rank ` { x } ) } )
95	62, 92, 94	mpd3an23 1317	. . . . . 6 |- ( Lim ( rank ` A ) -> ( cf ` ( rank ` A ) ) ~<_ { w | E. x e. A w = ( rank ` { x } ) } )
96		fveq2 5802	. . . . . . . . . . 11 |- ( y = A -> ( rank ` y ) = ( rank ` A ) )
97	96	fveq2d 5806	. . . . . . . . . 10 |- ( y = A -> ( cf ` ( rank ` y ) ) = ( cf ` ( rank ` A ) ) )
98		breq12 4408	. . . . . . . . . 10 |- ( ( y = A /\ ( cf ` ( rank ` y ) ) = ( cf ` ( rank ` A ) ) ) -> ( y ~< ( cf ` ( rank ` y ) ) <-> A ~< ( cf ` ( rank ` A ) ) ) )
99	97, 98	mpdan 668	. . . . . . . . 9 |- ( y = A -> ( y ~< ( cf ` ( rank ` y ) ) <-> A ~< ( cf ` ( rank ` A ) ) ) )
100		rexeq 3024	. . . . . . . . . . 11 |- ( y = A -> ( E. x e. y w = ( rank ` { x } ) <-> E. x e. A w = ( rank ` { x } ) ) )
101	100	abbidv 2590	. . . . . . . . . 10 |- ( y = A -> { w | E. x e. y w = ( rank ` { x } ) } = { w | E. x e. A w = ( rank ` { x } ) } )
102		breq12 4408	. . . . . . . . . 10 |- ( ( { w | E. x e. y w = ( rank ` { x } ) } = { w | E. x e. A w = ( rank ` { x } ) } /\ y = A ) -> ( { w | E. x e. y w = ( rank ` { x } ) } ~<_ y <-> { w | E. x e. A w = ( rank ` { x } ) } ~<_ A ) )
103	101, 102	mpancom 669	. . . . . . . . 9 |- ( y = A -> ( { w | E. x e. y w = ( rank ` { x } ) } ~<_ y <-> { w | E. x e. A w = ( rank ` { x } ) } ~<_ A ) )
104	99, 103	imbi12d 320	. . . . . . . 8 |- ( y = A -> ( ( y ~< ( cf ` ( rank ` y ) ) -> { w | E. x e. y w = ( rank ` { x } ) } ~<_ y ) <-> ( A ~< ( cf ` ( rank ` A ) ) -> { w | E. x e. A w = ( rank ` { x } ) } ~<_ A ) ) )
105		eqid 2454	. . . . . . . . . 10 |- ( x e. y |-> ( rank ` { x } ) ) = ( x e. y |-> ( rank ` { x } ) )
106	105	rnmpt 5196	. . . . . . . . 9 |- ran ( x e. y |-> ( rank ` { x } ) ) = { w | E. x e. y w = ( rank ` { x } ) }
107		cfon 8538	. . . . . . . . . . 11 |- ( cf ` ( rank ` y ) ) e. On
108		sdomdom 7450	. . . . . . . . . . 11 |- ( y ~< ( cf ` ( rank ` y ) ) -> y ~<_ ( cf ` ( rank ` y ) ) )
109		ondomen 8321	. . . . . . . . . . 11 |- ( ( ( cf ` ( rank ` y ) ) e. On /\ y ~<_ ( cf ` ( rank ` y ) ) ) -> y e. dom card )
110	107, 108, 109	sylancr 663	. . . . . . . . . 10 |- ( y ~< ( cf ` ( rank ` y ) ) -> y e. dom card )
111		fvex 5812	. . . . . . . . . . . 12 |- ( rank ` { x } ) e. _V
112	111, 105	fnmpti 5650	. . . . . . . . . . 11 |- ( x e. y |-> ( rank ` { x } ) ) Fn y
113		dffn4 5737	. . . . . . . . . . 11 |- ( ( x e. y |-> ( rank ` { x } ) ) Fn y <-> ( x e. y |-> ( rank ` { x } ) ) : y -onto-> ran ( x e. y |-> ( rank ` { x } ) ) )
114	112, 113	mpbi 208	. . . . . . . . . 10 |- ( x e. y |-> ( rank ` { x } ) ) : y -onto-> ran ( x e. y |-> ( rank ` { x } ) )
115		fodomnum 8341	. . . . . . . . . 10 |- ( y e. dom card -> ( ( x e. y |-> ( rank ` { x } ) ) : y -onto-> ran ( x e. y |-> ( rank ` { x } ) ) -> ran ( x e. y |-> ( rank ` { x } ) ) ~<_ y ) )
116	110, 114, 115	mpisyl 18	. . . . . . . . 9 |- ( y ~< ( cf ` ( rank ` y ) ) -> ran ( x e. y |-> ( rank ` { x } ) ) ~<_ y )
117	106, 116	syl5eqbrr 4437	. . . . . . . 8 |- ( y ~< ( cf ` ( rank ` y ) ) -> { w | E. x e. y w = ( rank ` { x } ) } ~<_ y )
118	104, 117	vtoclg 3136	. . . . . . 7 |- ( A e. U. ( R1 " On ) -> ( A ~< ( cf ` ( rank ` A ) ) -> { w | E. x e. A w = ( rank ` { x } ) } ~<_ A ) )
119	47, 118	syl 16	. . . . . 6 |- ( Lim ( rank ` A ) -> ( A ~< ( cf ` ( rank ` A ) ) -> { w | E. x e. A w = ( rank ` { x } ) } ~<_ A ) )
120		domtr 7475	. . . . . . 7 |- ( ( ( cf ` ( rank ` A ) ) ~<_ { w | E. x e. A w = ( rank ` { x } ) } /\ { w | E. x e. A w = ( rank ` { x } ) } ~<_ A ) -> ( cf ` ( rank ` A ) ) ~<_ A )
121	120, 41	syl 16	. . . . . 6 |- ( ( ( cf ` ( rank ` A ) ) ~<_ { w | E. x e. A w = ( rank ` { x } ) } /\ { w | E. x e. A w = ( rank ` { x } ) } ~<_ A ) -> -. A ~< ( cf ` ( rank ` A ) ) )
122	95, 119, 121	syl6an 545	. . . . 5 |- ( Lim ( rank ` A ) -> ( A ~< ( cf ` ( rank ` A ) ) -> -. A ~< ( cf ` ( rank ` A ) ) ) )
123	122	pm2.01d 169	. . . 4 |- ( Lim ( rank ` A ) -> -. A ~< ( cf ` ( rank ` A ) ) )
124	123	adantl 466	. . 3 |- ( ( ( rank ` A ) e. _V /\ Lim ( rank ` A ) ) -> -. A ~< ( cf ` ( rank ` A ) ) )
125	9, 42, 124	3jaoi 1282	. 2 |- ( ( ( rank ` A ) = (/) \/ E. x e. On ( rank ` A ) = suc x \/ ( ( rank ` A ) e. _V /\ Lim ( rank ` A ) ) ) -> -. A ~< ( cf ` ( rank ` A ) ) )
126	3, 125	ax-mp 5	1 |- -. A ~< ( cf ` ( rank ` A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   \/ w3o 964   = wceq 1370   e. wcel 1758   {cab 2439   =/= wne 2648   E.wrex 2800   _Vcvv 3078   C_ wss 3439   (/)c0 3748   {csn 3988   U.cuni 4202   U_ciun 4282   class class class wbr 4403   |-> cmpt 4461   Oncon0 4830   Lim wlim 4831   suc csuc 4832   dom cdm 4951   ran crn 4952   "cima 4954   Fn wfn 5524   -onto->wfo 5527   ` cfv 5529   1oc1o 7026   ~<_ cdom 7421   ~< csdm 7422   R1cr1 8083   rankcrnk 8084   cardccrd 8219   cfccf 8221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-r1 8085  df-rank 8086  df-card 8223  df-cf 8225  df-acn 8226

This theorem is referenced by:  inatsk  9059  grur1  9101