Theorem iscusp2 21366

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 21363	. 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 5891	. . . 4 |- ( Fil ` B ) = ( Fil ` ( Base ` W ) )
4		iscusp2.2	. . . . . . 7 |- U = (UnifSt ` W )
5	4	fveq2i 5891	. . . . . 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 6325	. . . . . 6 |- ( J fLim c ) = ( ( TopOpen ` W ) fLim c )
9	8	neeq1i 2700	. . . . 5 |- ( ( J fLim c ) =/= (/) <-> ( ( TopOpen ` W ) fLim c ) =/= (/) )
10	6, 9	imbi12i 332	. . . 4 |- ( ( c e. (CauFilu ` U ) -> ( J fLim c ) =/= (/) ) <-> ( c e. (CauFilu ` (UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) )
11	3, 10	raleqbii 2845	. . 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 705	. 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 260	1 |- ( W e. CUnifSp <-> ( W e. UnifSp /\ A. c e. ( Fil ` B ) ( c e. (CauFilu ` U ) -> ( J fLim c ) =/= (/) ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   =/= wne 2633   A.wral 2749   (/)c0 3743   ` cfv 5601  (class class class)co 6315   Basecbs 15170   TopOpenctopn 15369   Filcfil 20909   fLim cflim 20998  UnifStcuss 21317  UnifSpcusp 21318  CauFil_uccfilu 21350  CUnifSpccusp 21361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-iota 5565  df-fv 5609  df-ov 6318  df-cusp 21362

This theorem is referenced by:  cmetcusp1  22369