Theorem isufil 21707

Description: The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Assertion

Ref	Expression
isufil	|- ( F e. ( UFil ` X ) <-> ( F e. ( Fil ` X ) /\ A. x e. ~P X ( x e. F \/ ( X \ x ) e. F ) ) )

Distinct variable groups:   x, F   x, X

Proof of Theorem isufil

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

Step	Hyp	Ref	Expression
1		df-ufil 21705	. 2 |- UFil = ( y e. _V |-> { z e. ( Fil ` y ) | A. x e. ~P y ( x e. z \/ ( y \ x ) e. z ) } )
2		pweq 4161	. . . 4 |- ( y = X -> ~P y = ~P X )
3	2	adantr 481	. . 3 |- ( ( y = X /\ z = F ) -> ~P y = ~P X )
4		eleq2 2690	. . . . 5 |- ( z = F -> ( x e. z <-> x e. F ) )
5	4	adantl 482	. . . 4 |- ( ( y = X /\ z = F ) -> ( x e. z <-> x e. F ) )
6		difeq1 3721	. . . . 5 |- ( y = X -> ( y \ x ) = ( X \ x ) )
7		eleq12 2691	. . . . 5 |- ( ( ( y \ x ) = ( X \ x ) /\ z = F ) -> ( ( y \ x ) e. z <-> ( X \ x ) e. F ) )
8	6, 7	sylan 488	. . . 4 |- ( ( y = X /\ z = F ) -> ( ( y \ x ) e. z <-> ( X \ x ) e. F ) )
9	5, 8	orbi12d 746	. . 3 |- ( ( y = X /\ z = F ) -> ( ( x e. z \/ ( y \ x ) e. z ) <-> ( x e. F \/ ( X \ x ) e. F ) ) )
10	3, 9	raleqbidv 3152	. 2 |- ( ( y = X /\ z = F ) -> ( A. x e. ~P y ( x e. z \/ ( y \ x ) e. z ) <-> A. x e. ~P X ( x e. F \/ ( X \ x ) e. F ) ) )
11		fveq2 6191	. 2 |- ( y = X -> ( Fil ` y ) = ( Fil ` X ) )
12		fvex 6201	. 2 |- ( Fil ` y ) e. _V
13		elfvdm 6220	. 2 |- ( F e. ( Fil ` X ) -> X e. dom Fil )
14	1, 10, 11, 12, 13	elmptrab2 21632	1 |- ( F e. ( UFil ` X ) <-> ( F e. ( Fil ` X ) /\ A. x e. ~P X ( x e. F \/ ( X \ x ) e. F ) ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   ~Pcpw 4158   dom cdm 5114   ` cfv 5888   Filcfil 21649   UFilcufil 21703

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-ufil 21705

This theorem is referenced by:  ufilfil  21708  ufilss  21709  isufil2  21712  trufil  21714  fixufil  21726  fin1aufil  21736