Theorem iscusp 22103

Description: The predicate " W is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.)

Assertion

Ref	Expression
iscusp	|- ( W e. CUnifSp <-> ( W e. UnifSp /\ A. c e. ( Fil ` ( Base ` W ) ) ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) )

Distinct variable group:   W, c

Proof of Theorem iscusp

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
2	1	fveq2d 6195	. . 3 |- ( w = W -> ( Fil ` ( Base ` w ) ) = ( Fil ` ( Base ` W ) ) )
3		fveq2 6191	. . . . . 6 |- ( w = W -> (UnifSt ` w ) = (UnifSt ` W ) )
4	3	fveq2d 6195	. . . . 5 |- ( w = W -> (CauFilu ` (UnifSt ` w ) ) = (CauFilu ` (UnifSt ` W ) ) )
5	4	eleq2d 2687	. . . 4 |- ( w = W -> ( c e. (CauFilu ` (UnifSt ` w ) ) <-> c e. (CauFilu ` (UnifSt ` W ) ) ) )
6		fveq2 6191	. . . . . 6 |- ( w = W -> ( TopOpen ` w ) = ( TopOpen ` W ) )
7	6	oveq1d 6665	. . . . 5 |- ( w = W -> ( ( TopOpen ` w ) fLim c ) = ( ( TopOpen ` W ) fLim c ) )
8	7	neeq1d 2853	. . . 4 |- ( w = W -> ( ( ( TopOpen ` w ) fLim c ) =/= (/) <-> ( ( TopOpen ` W ) fLim c ) =/= (/) ) )
9	5, 8	imbi12d 334	. . 3 |- ( w = W -> ( ( c e. (CauFilu ` (UnifSt ` w ) ) -> ( ( TopOpen ` w ) fLim c ) =/= (/) ) <-> ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) )
10	2, 9	raleqbidv 3152	. 2 |- ( w = W -> ( A. c e. ( Fil ` ( Base ` w ) ) ( c e. (CauFilu ` (UnifSt ` w ) ) -> ( ( TopOpen ` w ) fLim c ) =/= (/) ) <-> A. c e. ( Fil ` ( Base ` W ) ) ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) )
11		df-cusp 22102	. 2 |- CUnifSp = { w e. UnifSp | A. c e. ( Fil ` ( Base ` w ) ) ( c e. (CauFilu ` (UnifSt ` w ) ) -> ( ( TopOpen ` w ) fLim c ) =/= (/) ) }
12	10, 11	elrab2 3366	1 |- ( W e. CUnifSp <-> ( W e. UnifSp /\ A. c e. ( Fil ` ( Base ` W ) ) ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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:  cuspusp  22104  cuspcvg  22105  iscusp2  22106  cmetcusp  23150