Theorem cmetcvg 23083

Description: The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.)

Hypothesis

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

Assertion

Ref	Expression
cmetcvg	|- ( ( D e. ( CMet ` X ) /\ F e. (CauFil ` D ) ) -> ( J fLim F ) =/= (/) )

Proof of Theorem cmetcvg

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscmet.1	. . . 4 |- J = ( MetOpen ` D )
2	1	iscmet 23082	. . 3 |- ( D e. ( CMet ` X ) <-> ( D e. ( Met ` X ) /\ A. f e. (CauFil ` D ) ( J fLim f ) =/= (/) ) )
3	2	simprbi 480	. 2 |- ( D e. ( CMet ` X ) -> A. f e. (CauFil ` D ) ( J fLim f ) =/= (/) )
4		oveq2 6658	. . . 4 |- ( f = F -> ( J fLim f ) = ( J fLim F ) )
5	4	neeq1d 2853	. . 3 |- ( f = F -> ( ( J fLim f ) =/= (/) <-> ( J fLim F ) =/= (/) ) )
6	5	rspccva 3308	. 2 |- ( ( A. f e. (CauFil ` D ) ( J fLim f ) =/= (/) /\ F e. (CauFil ` D ) ) -> ( J fLim F ) =/= (/) )
7	3, 6	sylan 488	1 |- ( ( D e. ( CMet ` X ) /\ F e. (CauFil ` D ) ) -> ( J fLim F ) =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)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:  cmetcaulem  23086  cmetss  23113  cmetcusp  23150  minveclem4a  23201  fmcncfil  29977