Theorem cuspcvg 22105

Description: In a complete uniform space, any Cauchy filter C has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.)

Hypotheses

Ref	Expression
cuspcvg.1	|- B = ( Base ` W )
cuspcvg.2	|- J = ( TopOpen ` W )

Assertion

Ref	Expression
cuspcvg	|- ( ( W e. CUnifSp /\ C e. (CauFilu ` (UnifSt ` W ) ) /\ C e. ( Fil ` B ) ) -> ( J fLim C ) =/= (/) )

Proof of Theorem cuspcvg

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . 5 |- ( c = C -> ( c e. (CauFilu ` (UnifSt ` W ) ) <-> C e. (CauFilu ` (UnifSt ` W ) ) ) )
2		cuspcvg.2	. . . . . . . . 9 |- J = ( TopOpen ` W )
3	2	eqcomi 2631	. . . . . . . 8 |- ( TopOpen ` W ) = J
4	3	a1i 11	. . . . . . 7 |- ( c = C -> ( TopOpen ` W ) = J )
5		id 22	. . . . . . 7 |- ( c = C -> c = C )
6	4, 5	oveq12d 6668	. . . . . 6 |- ( c = C -> ( ( TopOpen ` W ) fLim c ) = ( J fLim C ) )
7	6	neeq1d 2853	. . . . 5 |- ( c = C -> ( ( ( TopOpen ` W ) fLim c ) =/= (/) <-> ( J fLim C ) =/= (/) ) )
8	1, 7	imbi12d 334	. . . 4 |- ( c = C -> ( ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) <-> ( C e. (CauFilu ` (UnifSt ` W ) ) -> ( J fLim C ) =/= (/) ) ) )
9		iscusp 22103	. . . . . 6 |- ( W e. CUnifSp <-> ( W e. UnifSp /\ A. c e. ( Fil ` ( Base ` W ) ) ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) )
10	9	simprbi 480	. . . . 5 |- ( W e. CUnifSp -> A. c e. ( Fil ` ( Base ` W ) ) ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) )
11	10	adantr 481	. . . 4 |- ( ( W e. CUnifSp /\ C e. ( Fil ` B ) ) -> A. c e. ( Fil ` ( Base ` W ) ) ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) )
12		simpr 477	. . . . 5 |- ( ( W e. CUnifSp /\ C e. ( Fil ` B ) ) -> C e. ( Fil ` B ) )
13		cuspcvg.1	. . . . . 6 |- B = ( Base ` W )
14	13	fveq2i 6194	. . . . 5 |- ( Fil ` B ) = ( Fil ` ( Base ` W ) )
15	12, 14	syl6eleq 2711	. . . 4 |- ( ( W e. CUnifSp /\ C e. ( Fil ` B ) ) -> C e. ( Fil ` ( Base ` W ) ) )
16	8, 11, 15	rspcdva 3316	. . 3 |- ( ( W e. CUnifSp /\ C e. ( Fil ` B ) ) -> ( C e. (CauFilu ` (UnifSt ` W ) ) -> ( J fLim C ) =/= (/) ) )
17	16	3impia 1261	. 2 |- ( ( W e. CUnifSp /\ C e. ( Fil ` B ) /\ C e. (CauFilu ` (UnifSt ` W ) ) ) -> ( J fLim C ) =/= (/) )
18	17	3com23 1271	1 |- ( ( W e. CUnifSp /\ C e. (CauFilu ` (UnifSt ` W ) ) /\ C e. ( Fil ` B ) ) -> ( J fLim C ) =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Basecbs 15857   TopOpenctopn 16082   Filcfil 21649   fLim cflim 21738  UnifStcuss 22057  UnifSpcusp 22058  CauFil_uccfilu 22090  CUnifSpccusp 22101

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-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-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-iota 5851  df-fv 5896  df-ov 6653  df-cusp 22102

This theorem is referenced by:  cnextucn  22107