Theorem iscusp2 20931

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 20928	. 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 5875	. . . 4 |- ( Fil ` B ) = ( Fil ` ( Base ` W ) )
4		iscusp2.2	. . . . . . 7 |- U = (UnifSt ` W )
5	4	fveq2i 5875	. . . . . 6 |- (CauFilu ` U ) = (CauFilu ` (UnifSt ` W ) )
6	5	eleq2i 2535	. . . . 5 |- ( c e. (CauFilu ` U ) <-> c e. (CauFilu ` (UnifSt ` W ) ) )
7		iscusp2.3	. . . . . . 7 |- J = ( TopOpen ` W )
8	7	oveq1i 6306	. . . . . 6 |- ( J fLim c ) = ( ( TopOpen ` W ) fLim c )
9	8	neeq1i 2742	. . . . 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 2902	. . 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 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   (/)c0 3793   ` cfv 5594  (class class class)co 6296   Basecbs 14644   TopOpenctopn 14839   Filcfil 20472   fLim cflim 20561  UnifStcuss 20882  UnifSpcusp 20883  CauFil_uccfilu 20915  CUnifSpccusp 20926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-cusp 20927

This theorem is referenced by:  cmetcusp1OLD  21917  cmetcusp1  21918