Theorem cldval 8942

Description: The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.)

Hypothesis

Ref	Expression
cldval.1	|- X = U.J

Assertion

Ref	Expression
cldval	|- (J e. Top -> (Clsd` J) = {x | (x C_ X /\ (X \ x) e. J)})

Distinct variable groups:   x,J   x,X

Proof of Theorem cldval

Step	Hyp	Ref	Expression
1		uniexg 3795	. . . 4 |- (J e. Top -> U.J e. _V)
2		cldval.1	. . . 4 |- X = U.J
3	1, 2	syl5eqel 1975	. . 3 |- (J e. Top -> X e. _V)
4		abssexg 3490	. . 3 |- (X e. _V -> {x | (x C_ X /\ (X \ x) e. J)} e. _V)
5	3, 4	syl 12	. 2 |- (J e. Top -> {x | (x C_ X /\ (X \ x) e. J)} e. _V)
6		unieq 3185	. . . . . . 7 |- (z = J -> U.z = U.J)
7	6, 2	syl6eqr 1946	. . . . . 6 |- (z = J -> U.z = X)
8	7	sseq2d 2645	. . . . 5 |- (z = J -> (x C_ U.z <-> x C_ X))
9	7	difeq1d 2725	. . . . . 6 |- (z = J -> (U.z \ x) = (X \ x))
10		id 73	. . . . . 6 |- (z = J -> z = J)
11	9, 10	eleq12d 1965	. . . . 5 |- (z = J -> ((U.z \ x) e. z <-> (X \ x) e. J))
12	8, 11	anbi12d 690	. . . 4 |- (z = J -> ((x C_ U.z /\ (U.z \ x) e. z) <-> (x C_ X /\ (X \ x) e. J)))
13	12	abbidv 2008	. . 3 |- (z = J -> {x | (x C_ U.z /\ (U.z \ x) e. z)} = {x | (x C_ X /\ (X \ x) e. J)})
14		df-cld 8939	. . 3 |- Clsd = {<.z, w>. | (z e. Top /\ w = {x | (x C_ U.z /\ (U.z \ x) e. z)})}
15	13, 14	fvopab4g 4742	. 2 |- ((J e. Top /\ {x | (x C_ X /\ (X \ x) e. J)} e. _V) -> (Clsd` J) = {x | (x C_ X /\ (X \ x) e. J)})
16	5, 15	mpdan 768	1 |- (J e. Top -> (Clsd` J) = {x | (x C_ X /\ (X \ x) e. J)})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   \ cdif 2590   C_ wss 2593  U.cuni 3177  ` cfv 3998  Topctop 8857  Clsdccld 8936

This theorem is referenced by:  iscld 8945  clindistop 14962  singempcon 14965

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-13 1311  ax-14 1312  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  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-cld 8939

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