Theorem fbflim 21780

Description: A condition for a filter to converge to a point involving one of its bases. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
fbflim.3	|- F = ( X filGen B )

Assertion

Ref	Expression
fbflim	|- ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) -> ( A e. ( J fLim F ) <-> ( A e. X /\ A. x e. J ( A e. x -> E. y e. B y C_ x ) ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, J, y   x, X, y   x, F, y

Proof of Theorem fbflim

Step	Hyp	Ref	Expression
1		fbflim.3	. . . 4 |- F = ( X filGen B )
2		fgcl 21682	. . . 4 |- ( B e. ( fBas ` X ) -> ( X filGen B ) e. ( Fil ` X ) )
3	1, 2	syl5eqel 2705	. . 3 |- ( B e. ( fBas ` X ) -> F e. ( Fil ` X ) )
4		flimopn 21779	. . 3 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> ( A e. ( J fLim F ) <-> ( A e. X /\ A. x e. J ( A e. x -> x e. F ) ) ) )
5	3, 4	sylan2 491	. 2 |- ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) -> ( A e. ( J fLim F ) <-> ( A e. X /\ A. x e. J ( A e. x -> x e. F ) ) ) )
6	1	eleq2i 2693	. . . . . . 7 |- ( x e. F <-> x e. ( X filGen B ) )
7		elfg 21675	. . . . . . . 8 |- ( B e. ( fBas ` X ) -> ( x e. ( X filGen B ) <-> ( x C_ X /\ E. y e. B y C_ x ) ) )
8	7	ad3antlr 767	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) /\ A e. X ) /\ x e. J ) -> ( x e. ( X filGen B ) <-> ( x C_ X /\ E. y e. B y C_ x ) ) )
9	6, 8	syl5bb 272	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) /\ A e. X ) /\ x e. J ) -> ( x e. F <-> ( x C_ X /\ E. y e. B y C_ x ) ) )
10		simpll 790	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) /\ A e. X ) -> J e. (TopOn ` X ) )
11		toponss 20731	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ x e. J ) -> x C_ X )
12	10, 11	sylan 488	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) /\ A e. X ) /\ x e. J ) -> x C_ X )
13	12	biantrurd 529	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) /\ A e. X ) /\ x e. J ) -> ( E. y e. B y C_ x <-> ( x C_ X /\ E. y e. B y C_ x ) ) )
14	9, 13	bitr4d 271	. . . . 5 |- ( ( ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) /\ A e. X ) /\ x e. J ) -> ( x e. F <-> E. y e. B y C_ x ) )
15	14	imbi2d 330	. . . 4 |- ( ( ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) /\ A e. X ) /\ x e. J ) -> ( ( A e. x -> x e. F ) <-> ( A e. x -> E. y e. B y C_ x ) ) )
16	15	ralbidva 2985	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) /\ A e. X ) -> ( A. x e. J ( A e. x -> x e. F ) <-> A. x e. J ( A e. x -> E. y e. B y C_ x ) ) )
17	16	pm5.32da 673	. 2 |- ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) -> ( ( A e. X /\ A. x e. J ( A e. x -> x e. F ) ) <-> ( A e. X /\ A. x e. J ( A e. x -> E. y e. B y C_ x ) ) ) )
18	5, 17	bitrd 268	1 |- ( ( J e. (TopOn ` X ) /\ B e. ( fBas ` X ) ) -> ( A e. ( J fLim F ) <-> ( A e. X /\ A. x e. J ( A e. x -> E. y e. B y C_ x ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ` cfv 5888  (class class class)co 6650   fBascfbas 19734   filGencfg 19735  TopOnctopon 20715   Filcfil 21649   fLim cflim 21738

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-ntr 20824  df-nei 20902  df-fil 21650  df-flim 21743

This theorem is referenced by:  fbflim2  21781