Theorem filnet 32377

Description: A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Assertion

Ref	Expression
filnet	|- ( F e. ( Fil ` X ) -> E. d e. DirRel E. f ( f : dom d --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` d ) ) ) )

Distinct variable groups:   f, d, F   X, d, f

Proof of Theorem filnet

Dummy variables x y n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- U_ n e. F ( { n } X. n ) = U_ n e. F ( { n } X. n )
2		eqid 2622	. 2 |- { <. x , y >. | ( ( x e. U_ n e. F ( { n } X. n ) /\ y e. U_ n e. F ( { n } X. n ) ) /\ ( 1st ` y ) C_ ( 1st ` x ) ) } = { <. x , y >. | ( ( x e. U_ n e. F ( { n } X. n ) /\ y e. U_ n e. F ( { n } X. n ) ) /\ ( 1st ` y ) C_ ( 1st ` x ) ) }
3	1, 2	filnetlem4 32376	1 |- ( F e. ( Fil ` X ) -> E. d e. DirRel E. f ( f : dom d --> X /\ F = ( ( X FilMap f ) ` ran ( tail ` d ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   C_ wss 3574   {csn 4177   U_ciun 4520   {copab 4712   X. cxp 5112   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   DirRelcdir 17228   tailctail 17229   Filcfil 21649   FilMap cfm 21737

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-1st 7168  df-2nd 7169  df-dir 17230  df-tail 17231  df-fbas 19743  df-fg 19744  df-fil 21650  df-fm 21742

This theorem is referenced by: (None)