Theorem isfil 21651

Description: The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)

Assertion

Ref	Expression
isfil	|- ( F e. ( Fil ` X ) <-> ( F e. ( fBas ` X ) /\ A. x e. ~P X ( ( F i^i ~P x ) =/= (/) -> x e. F ) ) )

Distinct variable groups:   x, F   x, X

Proof of Theorem isfil

Dummy variables f z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fil 21650	. 2 |- Fil = ( z e. _V |-> { f e. ( fBas ` z ) | A. x e. ~P z ( ( f i^i ~P x ) =/= (/) -> x e. f ) } )
2		pweq 4161	. . . 4 |- ( z = X -> ~P z = ~P X )
3	2	adantr 481	. . 3 |- ( ( z = X /\ f = F ) -> ~P z = ~P X )
4		ineq1 3807	. . . . . 6 |- ( f = F -> ( f i^i ~P x ) = ( F i^i ~P x ) )
5	4	neeq1d 2853	. . . . 5 |- ( f = F -> ( ( f i^i ~P x ) =/= (/) <-> ( F i^i ~P x ) =/= (/) ) )
6		eleq2 2690	. . . . 5 |- ( f = F -> ( x e. f <-> x e. F ) )
7	5, 6	imbi12d 334	. . . 4 |- ( f = F -> ( ( ( f i^i ~P x ) =/= (/) -> x e. f ) <-> ( ( F i^i ~P x ) =/= (/) -> x e. F ) ) )
8	7	adantl 482	. . 3 |- ( ( z = X /\ f = F ) -> ( ( ( f i^i ~P x ) =/= (/) -> x e. f ) <-> ( ( F i^i ~P x ) =/= (/) -> x e. F ) ) )
9	3, 8	raleqbidv 3152	. 2 |- ( ( z = X /\ f = F ) -> ( A. x e. ~P z ( ( f i^i ~P x ) =/= (/) -> x e. f ) <-> A. x e. ~P X ( ( F i^i ~P x ) =/= (/) -> x e. F ) ) )
10		fveq2 6191	. 2 |- ( z = X -> ( fBas ` z ) = ( fBas ` X ) )
11		fvex 6201	. 2 |- ( fBas ` z ) e. _V
12		elfvdm 6220	. 2 |- ( F e. ( fBas ` X ) -> X e. dom fBas )
13	1, 9, 10, 11, 12	elmptrab2 21632	1 |- ( F e. ( Fil ` X ) <-> ( F e. ( fBas ` X ) /\ A. x e. ~P X ( ( F i^i ~P x ) =/= (/) -> x e. F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   dom cdm 5114   ` 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-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-fv 5896  df-fil 21650

This theorem is referenced by:  filfbas  21652  filss  21657  isfil2  21660  ustfilxp  22016