Theorem extbas2 10292

Description: The base set of an extended filter. (Contributed by Jeff Hankins, 7-Sep-2009.)

Hypothesis

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

Assertion

Ref	Expression
extbas2	|- ((X C_ A /\ A e. B) -> U.(F u. {A}) = A)

Proof of Theorem extbas2

Step	Hyp	Ref	Expression
1		unisng 3194	. . . 4 |- (A e. B -> U.{A} = A)
2	1	uneq2d 2755	. . 3 |- (A e. B -> (X u. U.{A}) = (X u. A))
3		ssid 2634	. . . . . 6 |- A C_ A
4	3	jctr 315	. . . . 5 |- (X C_ A -> (X C_ A /\ A C_ A))
5		unss 2780	. . . . 5 |- ((X C_ A /\ A C_ A) <-> (X u. A) C_ A)
6	4, 5	sylib 215	. . . 4 |- (X C_ A -> (X u. A) C_ A)
7		ssun2 2768	. . . . 5 |- A C_ (X u. A)
8	7	a1i 8	. . . 4 |- (X C_ A -> A C_ (X u. A))
9	6, 8	eqssd 2633	. . 3 |- (X C_ A -> (X u. A) = A)
10	2, 9	sylan9eqr 1951	. 2 |- ((X C_ A /\ A e. B) -> (X u. U.{A}) = A)
11		uniun 3196	. . 3 |- U.(F u. {A}) = (U.F u. U.{A})
12		extbas.1	. . . 4 |- X = U.F
13	12	uneq1i 2751	. . 3 |- (X u. U.{A}) = (U.F u. U.{A})
14	11, 13	eqtr4i 1911	. 2 |- U.(F u. {A}) = (X u. U.{A})
15	10, 14	syl5eq 1940	1 |- ((X C_ A /\ A e. B) -> U.(F u. {A}) = A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   u. cun 2591   C_ wss 2593  {csn 3044  U.cuni 3177

This theorem is referenced by:  fmbas 10311  elfilmap 10312  cnpfillim 15589  fcluscomp 15621

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-rex 2110  df-v 2294  df-un 2600  df-in 2603  df-ss 2605  df-sn 3049  df-pr 3050  df-uni 3178

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