Theorem iscusp2 20012

Description: The predicate " W is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.)

Hypotheses

Ref	Expression
iscusp2.1	|- B = ( Base ` W )
iscusp2.2	|- U = (UnifSt ` W )
iscusp2.3	|- J = ( TopOpen ` W )

Assertion

Ref	Expression
iscusp2	|- ( W e. CUnifSp <-> ( W e. UnifSp /\ A. c e. ( Fil ` B ) ( c e. (CauFilu ` U ) -> ( J fLim c ) =/= (/) ) ) )

Distinct variable group:   W, c

Allowed substitution hints:   B( c)   U( c)   J( c)

Proof of Theorem iscusp2

Step	Hyp	Ref	Expression
1		iscusp 20009	. 2 |- ( W e. CUnifSp <-> ( W e. UnifSp /\ A. c e. ( Fil ` ( Base ` W ) ) ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) )
2		iscusp2.1	. . . . 5 |- B = ( Base ` W )
3	2	fveq2i 5805	. . . 4 |- ( Fil ` B ) = ( Fil ` ( Base ` W ) )
4		iscusp2.2	. . . . . . 7 |- U = (UnifSt ` W )
5	4	fveq2i 5805	. . . . . 6 |- (CauFilu ` U ) = (CauFilu ` (UnifSt ` W ) )
6	5	eleq2i 2532	. . . . 5 |- ( c e. (CauFilu ` U ) <-> c e. (CauFilu ` (UnifSt ` W ) ) )
7		iscusp2.3	. . . . . . 7 |- J = ( TopOpen ` W )
8	7	oveq1i 6213	. . . . . 6 |- ( J fLim c ) = ( ( TopOpen ` W ) fLim c )
9	8	neeq1i 2737	. . . . 5 |- ( ( J fLim c ) =/= (/) <-> ( ( TopOpen ` W ) fLim c ) =/= (/) )
10	6, 9	imbi12i 326	. . . 4 |- ( ( c e. (CauFilu ` U ) -> ( J fLim c ) =/= (/) ) <-> ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) )
11	3, 10	raleqbii 2848	. . 3 |- ( A. c e. ( Fil ` B ) ( c e. (CauFilu ` U ) -> ( J fLim c ) =/= (/) ) <-> A. c e. ( Fil ` ( Base ` W ) ) ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) )
12	11	anbi2i 694	. 2 |- ( ( W e. UnifSp /\ A. c e. ( Fil ` B ) ( c e. (CauFilu ` U ) -> ( J fLim c ) =/= (/) ) ) <-> ( W e. UnifSp /\ A. c e. ( Fil ` ( Base ` W ) ) ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) )
13	1, 12	bitr4i 252	1 |- ( W e. CUnifSp <-> ( W e. UnifSp /\ A. c e. ( Fil ` B ) ( c e. (CauFilu ` U ) -> ( J fLim c ) =/= (/) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2648   A.wral 2799   (/)c0 3748   ` cfv 5529  (class class class)co 6203   Basecbs 14295   TopOpenctopn 14482   Filcfil 19553   fLim cflim 19642  UnifStcuss 19963  UnifSpcusp 19964  CauFil_uccfilu 19996  CUnifSpccusp 20007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ov 6206  df-cusp 20008

This theorem is referenced by:  cmetcusp1OLD  20998  cmetcusp1  20999