Theorem isufil 20272

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 20270	. 2 |- UFil = ( y e. _V |-> { z e. ( Fil ` y ) | A. x e. ~P y ( x e. z \/ ( y \ x ) e. z ) } )
2		pweq 4019	. . . 4 |- ( y = X -> ~P y = ~P X )
3	2	adantr 465	. . 3 |- ( ( y = X /\ z = F ) -> ~P y = ~P X )
4		eleq2 2540	. . . . 5 |- ( z = F -> ( x e. z <-> x e. F ) )
5	4	adantl 466	. . . 4 |- ( ( y = X /\ z = F ) -> ( x e. z <-> x e. F ) )
6		difeq1 3620	. . . . 5 |- ( y = X -> ( y \ x ) = ( X \ x ) )
7		eleq12 2543	. . . . 5 |- ( ( ( y \ x ) = ( X \ x ) /\ z = F ) -> ( ( y \ x ) e. z <-> ( X \ x ) e. F ) )
8	6, 7	sylan 471	. . . 4 |- ( ( y = X /\ z = F ) -> ( ( y \ x ) e. z <-> ( X \ x ) e. F ) )
9	5, 8	orbi12d 709	. . 3 |- ( ( y = X /\ z = F ) -> ( ( x e. z \/ ( y \ x ) e. z ) <-> ( x e. F \/ ( X \ x ) e. F ) ) )
10	3, 9	raleqbidv 3077	. 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 5872	. 2 |- ( y = X -> ( Fil ` y ) = ( Fil ` X ) )
12		fvex 5882	. 2 |- ( Fil ` y ) e. _V
13		elfvdm 5898	. 2 |- ( F e. ( Fil ` X ) -> X e. dom Fil )
14	1, 10, 11, 12, 13	elmptrab2 20197	1 |- ( F e. ( UFil ` X ) <-> ( F e. ( Fil ` X ) /\ A. x e. ~P X ( x e. F \/ ( X \ x ) e. F ) ) )

Syntax hints:   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2817   _Vcvv 3118   \ cdif 3478   ~Pcpw 4016   dom cdm 5005   ` cfv 5594   Filcfil 20214   UFilcufil 20268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ufil 20270

This theorem is referenced by:  ufilfil  20273  ufilss  20274  isufil2  20277  trufil  20279  fixufil  20291  fin1aufil  20301