Theorem iscmet 23082

Description: The property " D is a complete metric." meaning all Cauchy filters converge to a point in the space. (Contributed by Mario Carneiro, 1-May-2014.) (Revised by Mario Carneiro, 13-Oct-2015.)

Hypothesis

Ref	Expression
iscmet.1	|- J = ( MetOpen ` D )

Assertion

Ref	Expression
iscmet	|- ( D e. ( CMet ` X ) <-> ( D e. ( Met ` X ) /\ A. f e. (CauFil ` D ) ( J fLim f ) =/= (/) ) )

Distinct variable groups:   D, f   f, J   f, X

Proof of Theorem iscmet

Dummy variables d x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6221	. 2 |- ( D e. ( CMet ` X ) -> X e. _V )
2		elfvex 6221	. . 3 |- ( D e. ( Met ` X ) -> X e. _V )
3	2	adantr 481	. 2 |- ( ( D e. ( Met ` X ) /\ A. f e. (CauFil ` D ) ( J fLim f ) =/= (/) ) -> X e. _V )
4		fveq2 6191	. . . . . 6 |- ( x = X -> ( Met ` x ) = ( Met ` X ) )
5		rabeq 3192	. . . . . 6 |- ( ( Met ` x ) = ( Met ` X ) -> { d e. ( Met ` x ) | A. f e. (CauFil ` d ) ( ( MetOpen ` d ) fLim f ) =/= (/) } = { d e. ( Met ` X ) | A. f e. (CauFil ` d ) ( ( MetOpen ` d ) fLim f ) =/= (/) } )
6	4, 5	syl 17	. . . . 5 |- ( x = X -> { d e. ( Met ` x ) | A. f e. (CauFil ` d ) ( ( MetOpen ` d ) fLim f ) =/= (/) } = { d e. ( Met ` X ) | A. f e. (CauFil ` d ) ( ( MetOpen ` d ) fLim f ) =/= (/) } )
7		df-cmet 23055	. . . . 5 |- CMet = ( x e. _V |-> { d e. ( Met ` x ) | A. f e. (CauFil ` d ) ( ( MetOpen ` d ) fLim f ) =/= (/) } )
8		fvex 6201	. . . . . 6 |- ( Met ` X ) e. _V
9	8	rabex 4813	. . . . 5 |- { d e. ( Met ` X ) | A. f e. (CauFil ` d ) ( ( MetOpen ` d ) fLim f ) =/= (/) } e. _V
10	6, 7, 9	fvmpt 6282	. . . 4 |- ( X e. _V -> ( CMet ` X ) = { d e. ( Met ` X ) | A. f e. (CauFil ` d ) ( ( MetOpen ` d ) fLim f ) =/= (/) } )
11	10	eleq2d 2687	. . 3 |- ( X e. _V -> ( D e. ( CMet ` X ) <-> D e. { d e. ( Met ` X ) | A. f e. (CauFil ` d ) ( ( MetOpen ` d ) fLim f ) =/= (/) } ) )
12		fveq2 6191	. . . . 5 |- ( d = D -> (CauFil ` d ) = (CauFil ` D ) )
13		fveq2 6191	. . . . . . . 8 |- ( d = D -> ( MetOpen ` d ) = ( MetOpen ` D ) )
14		iscmet.1	. . . . . . . 8 |- J = ( MetOpen ` D )
15	13, 14	syl6eqr 2674	. . . . . . 7 |- ( d = D -> ( MetOpen ` d ) = J )
16	15	oveq1d 6665	. . . . . 6 |- ( d = D -> ( ( MetOpen ` d ) fLim f ) = ( J fLim f ) )
17	16	neeq1d 2853	. . . . 5 |- ( d = D -> ( ( ( MetOpen ` d ) fLim f ) =/= (/) <-> ( J fLim f ) =/= (/) ) )
18	12, 17	raleqbidv 3152	. . . 4 |- ( d = D -> ( A. f e. (CauFil ` d ) ( ( MetOpen ` d ) fLim f ) =/= (/) <-> A. f e. (CauFil ` D ) ( J fLim f ) =/= (/) ) )
19	18	elrab 3363	. . 3 |- ( D e. { d e. ( Met ` X ) | A. f e. (CauFil ` d ) ( ( MetOpen ` d ) fLim f ) =/= (/) } <-> ( D e. ( Met ` X ) /\ A. f e. (CauFil ` D ) ( J fLim f ) =/= (/) ) )
20	11, 19	syl6bb 276	. 2 |- ( X e. _V -> ( D e. ( CMet ` X ) <-> ( D e. ( Met ` X ) /\ A. f e. (CauFil ` D ) ( J fLim f ) =/= (/) ) ) )
21	1, 3, 20	pm5.21nii 368	1 |- ( D e. ( CMet ` X ) <-> ( D e. ( Met ` X ) /\ A. f e. (CauFil ` D ) ( J fLim f ) =/= (/) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Metcme 19732   MetOpencmopn 19736   fLim cflim 21738  CauFilccfil 23050   CMetcms 23052

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

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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-cmet 23055

This theorem is referenced by:  cmetcvg  23083  cmetmet  23084  iscmet3  23091  cmetss  23113  equivcmet  23114  relcmpcmet  23115  cmetcusp1  23149