Theorem filunirn 21686

Description: Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
filunirn	|- ( F e. U. ran Fil <-> F e. ( Fil ` U. F ) )

Proof of Theorem filunirn

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

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . . 6 |- ( fBas ` y ) e. _V
2	1	rabex 4813	. . . . 5 |- { w e. ( fBas ` y ) | A. z e. ~P y ( ( w i^i ~P z ) =/= (/) -> z e. w ) } e. _V
3		df-fil 21650	. . . . 5 |- Fil = ( y e. _V |-> { w e. ( fBas ` y ) | A. z e. ~P y ( ( w i^i ~P z ) =/= (/) -> z e. w ) } )
4	2, 3	fnmpti 6022	. . . 4 |- Fil Fn _V
5		fnunirn 6511	. . . 4 |- ( Fil Fn _V -> ( F e. U. ran Fil <-> E. x e. _V F e. ( Fil ` x ) ) )
6	4, 5	ax-mp 5	. . 3 |- ( F e. U. ran Fil <-> E. x e. _V F e. ( Fil ` x ) )
7		filunibas 21685	. . . . . . 7 |- ( F e. ( Fil ` x ) -> U. F = x )
8	7	fveq2d 6195	. . . . . 6 |- ( F e. ( Fil ` x ) -> ( Fil ` U. F ) = ( Fil ` x ) )
9	8	eleq2d 2687	. . . . 5 |- ( F e. ( Fil ` x ) -> ( F e. ( Fil ` U. F ) <-> F e. ( Fil ` x ) ) )
10	9	ibir 257	. . . 4 |- ( F e. ( Fil ` x ) -> F e. ( Fil ` U. F ) )
11	10	rexlimivw 3029	. . 3 |- ( E. x e. _V F e. ( Fil ` x ) -> F e. ( Fil ` U. F ) )
12	6, 11	sylbi 207	. 2 |- ( F e. U. ran Fil -> F e. ( Fil ` U. F ) )
13		fvssunirn 6217	. . 3 |- ( Fil ` U. F ) C_ U. ran Fil
14	13	sseli 3599	. 2 |- ( F e. ( Fil ` U. F ) -> F e. U. ran Fil )
15	12, 14	impbii 199	1 |- ( F e. U. ran Fil <-> F e. ( Fil ` U. F ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   ran crn 5115   Fn wfn 5883   ` cfv 5888   fBascfbas 19734   Filcfil 21649

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-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-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-fv 5896  df-fbas 19743  df-fil 21650

This theorem is referenced by:  flimfil  21773  isfcls  21813