Theorem caublcls 23107

Description: The convergent point of a sequence of nested balls is in the closures of any of the balls (i.e. it is in the intersection of the closures). Indeed, it is the only point in the intersection because a metric space is Hausdorff, but we don't prove this here. (Contributed by Mario Carneiro, 21-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)

Hypotheses

Ref	Expression
caubl.2	|- ( ph -> D e. ( *Met ` X ) )
caubl.3	|- ( ph -> F : NN --> ( X X. RR+ ) )
caubl.4	|- ( ph -> A. n e. NN ( ( ball ` D ) ` ( F ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) )
caublcls.6	|- J = ( MetOpen ` D )

Assertion

Ref	Expression
caublcls	|- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> P e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( F ` A ) ) ) )

Distinct variable groups:   D, n   n, F   n, X

Allowed substitution hints:   ph( n)   A( n)   P( n)   J( n)

Proof of Theorem caublcls

Dummy variables k r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( ZZ>= ` A ) = ( ZZ>= ` A )
2		caubl.2	. . . 4 |- ( ph -> D e. ( *Met ` X ) )
3	2	3ad2ant1 1082	. . 3 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> D e. ( *Met ` X ) )
4		caublcls.6	. . . 4 |- J = ( MetOpen ` D )
5	4	mopntopon 22244	. . 3 |- ( D e. ( *Met ` X ) -> J e. (TopOn ` X ) )
6	3, 5	syl 17	. 2 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> J e. (TopOn ` X ) )
7		simp3 1063	. . 3 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> A e. NN )
8	7	nnzd 11481	. 2 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> A e. ZZ )
9		simp2 1062	. 2 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> ( 1st o. F ) ( ~~> t ` J ) P )
10		fveq2 6191	. . . . . . . . 9 |- ( r = A -> ( F ` r ) = ( F ` A ) )
11	10	fveq2d 6195	. . . . . . . 8 |- ( r = A -> ( ( ball ` D ) ` ( F ` r ) ) = ( ( ball ` D ) ` ( F ` A ) ) )
12	11	sseq1d 3632	. . . . . . 7 |- ( r = A -> ( ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) <-> ( ( ball ` D ) ` ( F ` A ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) )
13	12	imbi2d 330	. . . . . 6 |- ( r = A -> ( ( ( ph /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) <-> ( ( ph /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` A ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) ) )
14		fveq2 6191	. . . . . . . . 9 |- ( r = k -> ( F ` r ) = ( F ` k ) )
15	14	fveq2d 6195	. . . . . . . 8 |- ( r = k -> ( ( ball ` D ) ` ( F ` r ) ) = ( ( ball ` D ) ` ( F ` k ) ) )
16	15	sseq1d 3632	. . . . . . 7 |- ( r = k -> ( ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) <-> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) )
17	16	imbi2d 330	. . . . . 6 |- ( r = k -> ( ( ( ph /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) <-> ( ( ph /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) ) )
18		fveq2 6191	. . . . . . . . 9 |- ( r = ( k + 1 ) -> ( F ` r ) = ( F ` ( k + 1 ) ) )
19	18	fveq2d 6195	. . . . . . . 8 |- ( r = ( k + 1 ) -> ( ( ball ` D ) ` ( F ` r ) ) = ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) )
20	19	sseq1d 3632	. . . . . . 7 |- ( r = ( k + 1 ) -> ( ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) <-> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) )
21	20	imbi2d 330	. . . . . 6 |- ( r = ( k + 1 ) -> ( ( ( ph /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) <-> ( ( ph /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) ) )
22		ssid 3624	. . . . . . 7 |- ( ( ball ` D ) ` ( F ` A ) ) C_ ( ( ball ` D ) ` ( F ` A ) )
23	22	2a1i 12	. . . . . 6 |- ( A e. ZZ -> ( ( ph /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` A ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) )
24		caubl.4	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( ( ball ` D ) ` ( F ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) )
25		eluznn 11758	. . . . . . . . . . 11 |- ( ( A e. NN /\ k e. ( ZZ>= ` A ) ) -> k e. NN )
26		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( n = k -> ( n + 1 ) = ( k + 1 ) )
27	26	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( n = k -> ( F ` ( n + 1 ) ) = ( F ` ( k + 1 ) ) )
28	27	fveq2d 6195	. . . . . . . . . . . . 13 |- ( n = k -> ( ( ball ` D ) ` ( F ` ( n + 1 ) ) ) = ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) )
29		fveq2 6191	. . . . . . . . . . . . . 14 |- ( n = k -> ( F ` n ) = ( F ` k ) )
30	29	fveq2d 6195	. . . . . . . . . . . . 13 |- ( n = k -> ( ( ball ` D ) ` ( F ` n ) ) = ( ( ball ` D ) ` ( F ` k ) ) )
31	28, 30	sseq12d 3634	. . . . . . . . . . . 12 |- ( n = k -> ( ( ( ball ` D ) ` ( F ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) <-> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` k ) ) ) )
32	31	rspccva 3308	. . . . . . . . . . 11 |- ( ( A. n e. NN ( ( ball ` D ) ` ( F ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) /\ k e. NN ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` k ) ) )
33	24, 25, 32	syl2an 494	. . . . . . . . . 10 |- ( ( ph /\ ( A e. NN /\ k e. ( ZZ>= ` A ) ) ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` k ) ) )
34	33	anassrs 680	. . . . . . . . 9 |- ( ( ( ph /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` k ) ) )
35		sstr2 3610	. . . . . . . . 9 |- ( ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` k ) ) -> ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) )
36	34, 35	syl 17	. . . . . . . 8 |- ( ( ( ph /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) )
37	36	expcom 451	. . . . . . 7 |- ( k e. ( ZZ>= ` A ) -> ( ( ph /\ A e. NN ) -> ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) ) )
38	37	a2d 29	. . . . . 6 |- ( k e. ( ZZ>= ` A ) -> ( ( ( ph /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) -> ( ( ph /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) ) )
39	13, 17, 21, 17, 23, 38	uzind4 11746	. . . . 5 |- ( k e. ( ZZ>= ` A ) -> ( ( ph /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) ) )
40	39	impcom 446	. . . 4 |- ( ( ( ph /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) )
41	40	3adantl2 1218	. . 3 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` A ) ) )
42	3	adantr 481	. . . . 5 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> D e. ( *Met ` X ) )
43		simpl1 1064	. . . . . . . 8 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ph )
44		caubl.3	. . . . . . . 8 |- ( ph -> F : NN --> ( X X. RR+ ) )
45	43, 44	syl 17	. . . . . . 7 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> F : NN --> ( X X. RR+ ) )
46	25	3ad2antl3 1225	. . . . . . 7 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> k e. NN )
47	45, 46	ffvelrnd 6360	. . . . . 6 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( F ` k ) e. ( X X. RR+ ) )
48		xp1st 7198	. . . . . 6 |- ( ( F ` k ) e. ( X X. RR+ ) -> ( 1st ` ( F ` k ) ) e. X )
49	47, 48	syl 17	. . . . 5 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( 1st ` ( F ` k ) ) e. X )
50		xp2nd 7199	. . . . . 6 |- ( ( F ` k ) e. ( X X. RR+ ) -> ( 2nd ` ( F ` k ) ) e. RR+ )
51	47, 50	syl 17	. . . . 5 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( 2nd ` ( F ` k ) ) e. RR+ )
52		blcntr 22218	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ ( 1st ` ( F ` k ) ) e. X /\ ( 2nd ` ( F ` k ) ) e. RR+ ) -> ( 1st ` ( F ` k ) ) e. ( ( 1st ` ( F ` k ) ) ( ball ` D ) ( 2nd ` ( F ` k ) ) ) )
53	42, 49, 51, 52	syl3anc 1326	. . . 4 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( 1st ` ( F ` k ) ) e. ( ( 1st ` ( F ` k ) ) ( ball ` D ) ( 2nd ` ( F ` k ) ) ) )
54		fvco3 6275	. . . . 5 |- ( ( F : NN --> ( X X. RR+ ) /\ k e. NN ) -> ( ( 1st o. F ) ` k ) = ( 1st ` ( F ` k ) ) )
55	45, 46, 54	syl2anc 693	. . . 4 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( ( 1st o. F ) ` k ) = ( 1st ` ( F ` k ) ) )
56		1st2nd2 7205	. . . . . . 7 |- ( ( F ` k ) e. ( X X. RR+ ) -> ( F ` k ) = <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. )
57	47, 56	syl 17	. . . . . 6 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( F ` k ) = <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. )
58	57	fveq2d 6195	. . . . 5 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( ( ball ` D ) ` ( F ` k ) ) = ( ( ball ` D ) ` <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. ) )
59		df-ov 6653	. . . . 5 |- ( ( 1st ` ( F ` k ) ) ( ball ` D ) ( 2nd ` ( F ` k ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. )
60	58, 59	syl6eqr 2674	. . . 4 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( ( ball ` D ) ` ( F ` k ) ) = ( ( 1st ` ( F ` k ) ) ( ball ` D ) ( 2nd ` ( F ` k ) ) ) )
61	53, 55, 60	3eltr4d 2716	. . 3 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( ( 1st o. F ) ` k ) e. ( ( ball ` D ) ` ( F ` k ) ) )
62	41, 61	sseldd 3604	. 2 |- ( ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) /\ k e. ( ZZ>= ` A ) ) -> ( ( 1st o. F ) ` k ) e. ( ( ball ` D ) ` ( F ` A ) ) )
63	44	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ A e. NN ) -> ( F ` A ) e. ( X X. RR+ ) )
64	63	3adant2 1080	. . . . . 6 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> ( F ` A ) e. ( X X. RR+ ) )
65		1st2nd2 7205	. . . . . 6 |- ( ( F ` A ) e. ( X X. RR+ ) -> ( F ` A ) = <. ( 1st ` ( F ` A ) ) , ( 2nd ` ( F ` A ) ) >. )
66	64, 65	syl 17	. . . . 5 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> ( F ` A ) = <. ( 1st ` ( F ` A ) ) , ( 2nd ` ( F ` A ) ) >. )
67	66	fveq2d 6195	. . . 4 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` A ) ) = ( ( ball ` D ) ` <. ( 1st ` ( F ` A ) ) , ( 2nd ` ( F ` A ) ) >. ) )
68		df-ov 6653	. . . 4 |- ( ( 1st ` ( F ` A ) ) ( ball ` D ) ( 2nd ` ( F ` A ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( F ` A ) ) , ( 2nd ` ( F ` A ) ) >. )
69	67, 68	syl6eqr 2674	. . 3 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` A ) ) = ( ( 1st ` ( F ` A ) ) ( ball ` D ) ( 2nd ` ( F ` A ) ) ) )
70		xp1st 7198	. . . . 5 |- ( ( F ` A ) e. ( X X. RR+ ) -> ( 1st ` ( F ` A ) ) e. X )
71	64, 70	syl 17	. . . 4 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> ( 1st ` ( F ` A ) ) e. X )
72		xp2nd 7199	. . . . . 6 |- ( ( F ` A ) e. ( X X. RR+ ) -> ( 2nd ` ( F ` A ) ) e. RR+ )
73	64, 72	syl 17	. . . . 5 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> ( 2nd ` ( F ` A ) ) e. RR+ )
74	73	rpxrd 11873	. . . 4 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> ( 2nd ` ( F ` A ) ) e. RR* )
75		blssm 22223	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( 1st ` ( F ` A ) ) e. X /\ ( 2nd ` ( F ` A ) ) e. RR* ) -> ( ( 1st ` ( F ` A ) ) ( ball ` D ) ( 2nd ` ( F ` A ) ) ) C_ X )
76	3, 71, 74, 75	syl3anc 1326	. . 3 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> ( ( 1st ` ( F ` A ) ) ( ball ` D ) ( 2nd ` ( F ` A ) ) ) C_ X )
77	69, 76	eqsstrd 3639	. 2 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> ( ( ball ` D ) ` ( F ` A ) ) C_ X )
78	1, 6, 8, 9, 62, 77	lmcls 21106	1 |- ( ( ph /\ ( 1st o. F ) ( ~~> t ` J ) P /\ A e. NN ) -> P e. ( ( cls ` J ) ` ( ( ball ` D ) ` ( F ` A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   <.cop 4183   class class class wbr 4653   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   1c1 9937   + caddc 9939   RR*cxr 10073   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736  TopOnctopon 20715   clsccl 20822   ~~> tclm 21030

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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rmo 2920  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-iin 4523  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-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-lm 21033

This theorem is referenced by:  bcthlem3  23123  heiborlem8  33617