Theorem isufil 19618

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 19616	. 2 |- UFil = ( y e. _V |-> { z e. ( Fil ` y ) | A. x e. ~P y ( x e. z \/ ( y \ x ) e. z ) } )
2		pweq 3974	. . . 4 |- ( y = X -> ~P y = ~P X )
3	2	adantr 465	. . 3 |- ( ( y = X /\ z = F ) -> ~P y = ~P X )
4		eleq2 2527	. . . . 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 3578	. . . . 5 |- ( y = X -> ( y \ x ) = ( X \ x ) )
7		eleq12 2530	. . . . 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 3037	. 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 5802	. 2 |- ( y = X -> ( Fil ` y ) = ( Fil ` X ) )
12		fvex 5812	. 2 |- ( Fil ` y ) e. _V
13		elfvdm 5828	. 2 |- ( F e. ( Fil ` X ) -> X e. dom Fil )
14	1, 10, 11, 12, 13	elmptrab2 19543	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 1370   e. wcel 1758   A.wral 2799   _Vcvv 3078   \ cdif 3436   ~Pcpw 3971   dom cdm 4951   ` cfv 5529   Filcfil 19560   UFilcufil 19614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fv 5537  df-ufil 19616

This theorem is referenced by:  ufilfil  19619  ufilss  19620  isufil2  19623  trufil  19625  fixufil  19637  fin1aufil  19647