Theorem isufil 15564

Description: The property of being an ultrafilter.

Hypothesis

Ref	Expression
isufil.1	|- X = U.F

Assertion

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

Distinct variable groups:   x,F   x,X

Proof of Theorem isufil

Step	Hyp	Ref	Expression
1		unieq 3185	. . . . 5 |- (f = F -> U.f = U.F)
2		isufil.1	. . . . 5 |- X = U.F
3	1, 2	syl6eqr 1946	. . . 4 |- (f = F -> U.f = X)
4		pweq 3036	. . . 4 |- (U.f = X -> ~PU.f = ~PX)
5	3, 4	syl 12	. . 3 |- (f = F -> ~PU.f = ~PX)
6		eleq2 1958	. . . 4 |- (f = F -> (x e. f <-> x e. F))
7	3	difeq1d 2725	. . . . 5 |- (f = F -> (U.f \ x) = (X \ x))
8		id 73	. . . . 5 |- (f = F -> f = F)
9	7, 8	eleq12d 1965	. . . 4 |- (f = F -> ((U.f \ x) e. f <-> (X \ x) e. F))
10	6, 9	orbi12d 689	. . 3 |- (f = F -> ((x e. f \/ (U.f \ x) e. f) <-> (x e. F \/ (X \ x) e. F)))
11	5, 10	raleqbidv 2274	. 2 |- (f = F -> (A.x e. ~P U.f(x e. f \/ (U.f \ x) e. f) <-> A.x e. ~P X(x e. F \/ (X \ x) e. F)))
12		df-ufil 15563	. 2 |- UFil = {f e. Fil | A.x e. ~P U.f(x e. f \/ (U.f \ x) e. f)}
13	11, 12	elrab2 2416	1 |- (F e. UFil <-> (F e. Fil /\ A.x e. ~P X(x e. F \/ (X \ x) e. F)))

Syntax hints:   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   \ cdif 2590  ~Pcpw 3032  U.cuni 3177  Filcfil 10264  UFilcufil 15562

This theorem is referenced by:  isufil2 15565  ufilfil 15566  ufilss 15567  ufileu 15573  fixufil 15576

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-pw 3035  df-uni 3178  df-ufil 15563

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