Theorem caufval 23073

Description: The set of Cauchy sequences on a metric space. (Contributed by NM, 8-Sep-2006.) (Revised by Mario Carneiro, 5-Sep-2015.)

Assertion

Ref	Expression
caufval	|- ( D e. ( *Met ` X ) -> ( Cau ` D ) = { f e. ( X ^pm CC ) | A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) } )

Distinct variable groups:   f, k, x, D   f, X, k, x

Proof of Theorem caufval

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cau 23054	. . 3 |- Cau = ( d e. U. ran *Met |-> { f e. ( dom dom d ^pm CC ) | A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` d ) x ) } )
2	1	a1i 11	. 2 |- ( D e. ( *Met ` X ) -> Cau = ( d e. U. ran *Met |-> { f e. ( dom dom d ^pm CC ) | A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` d ) x ) } ) )
3		dmeq 5324	. . . . . 6 |- ( d = D -> dom d = dom D )
4	3	dmeqd 5326	. . . . 5 |- ( d = D -> dom dom d = dom dom D )
5		xmetf 22134	. . . . . . . 8 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
6		fdm 6051	. . . . . . . 8 |- ( D : ( X X. X ) --> RR* -> dom D = ( X X. X ) )
7	5, 6	syl 17	. . . . . . 7 |- ( D e. ( *Met ` X ) -> dom D = ( X X. X ) )
8	7	dmeqd 5326	. . . . . 6 |- ( D e. ( *Met ` X ) -> dom dom D = dom ( X X. X ) )
9		dmxpid 5345	. . . . . 6 |- dom ( X X. X ) = X
10	8, 9	syl6eq 2672	. . . . 5 |- ( D e. ( *Met ` X ) -> dom dom D = X )
11	4, 10	sylan9eqr 2678	. . . 4 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> dom dom d = X )
12	11	oveq1d 6665	. . 3 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> ( dom dom d ^pm CC ) = ( X ^pm CC ) )
13		simpr 477	. . . . . . . 8 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> d = D )
14	13	fveq2d 6195	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> ( ball ` d ) = ( ball ` D ) )
15	14	oveqd 6667	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> ( ( f ` k ) ( ball ` d ) x ) = ( ( f ` k ) ( ball ` D ) x ) )
16	15	feq3d 6032	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> ( ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` d ) x ) <-> ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) ) )
17	16	rexbidv 3052	. . . 4 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> ( E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` d ) x ) <-> E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) ) )
18	17	ralbidv 2986	. . 3 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> ( A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` d ) x ) <-> A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) ) )
19	12, 18	rabeqbidv 3195	. 2 |- ( ( D e. ( *Met ` X ) /\ d = D ) -> { f e. ( dom dom d ^pm CC ) | A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` d ) x ) } = { f e. ( X ^pm CC ) | A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) } )
20		fvssunirn 6217	. . 3 |- ( *Met ` X ) C_ U. ran *Met
21	20	sseli 3599	. 2 |- ( D e. ( *Met ` X ) -> D e. U. ran *Met )
22		ovex 6678	. . . 4 |- ( X ^pm CC ) e. _V
23	22	rabex 4813	. . 3 |- { f e. ( X ^pm CC ) | A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) } e. _V
24	23	a1i 11	. 2 |- ( D e. ( *Met ` X ) -> { f e. ( X ^pm CC ) | A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) } e. _V )
25	2, 19, 21, 24	fvmptd 6288	1 |- ( D e. ( *Met ` X ) -> ( Cau ` D ) = { f e. ( X ^pm CC ) | A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RR*cxr 10073   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   *Metcxmt 19731   ballcbl 19733   Caucca 23051

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-xr 10078  df-xmet 19739  df-cau 23054

This theorem is referenced by:  iscau  23074  equivcau  23098