Theorem cldval 7666

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 (_ X /\ (X \ x) e. J)})

Distinct variable groups:   x,J   x,X

Proof of Theorem cldval

Step	Hyp	Ref	Expression
1		uniexg 2871	. . . 4 |- (J e. Top -> U.J e. V)
2		cldval.1	. . . 4 |- X = U.J
3	1, 2	syl5eqel 1552	. . 3 |- (J e. Top -> X e. V)
4		abssexg 2747	. . 3 |- (X e. V -> {x | (x (_ X /\ (X \ x) e. J)} e. V)
5	3, 4	syl 10	. 2 |- (J e. Top -> {x | (x (_ X /\ (X \ x) e. J)} e. V)
6		unieq 2510	. . . . . . 7 |- (z = J -> U.z = U.J)
7	6, 2	syl6eqr 1525	. . . . . 6 |- (z = J -> U.z = X)
8	7	sseq2d 2089	. . . . 5 |- (z = J -> (x (_ U.z <-> x (_ X))
9	7	difeq1d 2158	. . . . . 6 |- (z = J -> (U.z \ x) = (X \ x))
10		id 59	. . . . . 6 |- (z = J -> z = J)
11	9, 10	eleq12d 1542	. . . . 5 |- (z = J -> ((U.z \ x) e. z <-> (X \ x) e. J))
12	8, 11	anbi12d 628	. . . 4 |- (z = J -> ((x (_ U.z /\ (U.z \ x) e. z) <-> (x (_ X /\ (X \ x) e. J)))
13	12	abbidv 1577	. . 3 |- (z = J -> {x | (x (_ U.z /\ (U.z \ x) e. z)} = {x | (x (_ X /\ (X \ x) e. J)})
14		df-cld 7663	. . 3 |- Clsd = {<.z, w>. | (z e. Top /\ w = {x | (x (_ U.z /\ (U.z \ x) e. z)})}
15	13, 14	fvopab4g 3779	. 2 |- ((J e. Top /\ {x | (x (_ X /\ (X \ x) e. J)} e. V) -> (Clsd` J) = {x | (x (_ X /\ (X \ x) e. J)})
16	5, 15	mpdan 704	1 |- (J e. Top -> (Clsd` J) = {x | (x (_ X /\ (X \ x) e. J)})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   \ cdif 2044   (_ wss 2047  U.cuni 2503  ` cfv 3182  Topctop 7588  Clsdccld 7660

This theorem is referenced by:  iscld 7669

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-cld 7663

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