Theorem cldval 7645

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 2868	. . . 4 |- (J e. Top -> U.J e. V)
2		cldval.1	. . . 4 |- X = U.J
3	1, 2	syl5eqel 1551	. . 3 |- (J e. Top -> X e. V)
4		abssexg 2744	. . 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 2507	. . . . . . 7 |- (z = J -> U.z = U.J)
7	6, 2	syl6eqr 1524	. . . . . 6 |- (z = J -> U.z = X)
8	7	sseq2d 2087	. . . . 5 |- (z = J -> (x (_ U.z <-> x (_ X))
9	7	difeq1d 2156	. . . . . 6 |- (z = J -> (U.z \ x) = (X \ x))
10		id 59	. . . . . 6 |- (z = J -> z = J)
11	9, 10	eleq12d 1541	. . . . 5 |- (z = J -> ((U.z \ x) e. z <-> (X \ x) e. J))
12	8, 11	anbi12d 627	. . . 4 |- (z = J -> ((x (_ U.z /\ (U.z \ x) e. z) <-> (x (_ X /\ (X \ x) e. J)))
13	12	abbidv 1576	. . 3 |- (z = J -> {x | (x (_ U.z /\ (U.z \ x) e. z)} = {x | (x (_ X /\ (X \ x) e. J)})
14		df-cld 7642	. . 3 |- Clsd = {<.z, w>. | (z e. Top /\ w = {x | (x (_ U.z /\ (U.z \ x) e. z)})}
15	13, 14	fvopab4g 3776	. 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 703	1 |- (J e. Top -> (Clsd` J) = {x | (x (_ X /\ (X \ x) e. J)})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  {cab 1463  Vcvv 1809   \ cdif 2042   (_ wss 2045  U.cuni 2500  ` cfv 3179  Topctop 7567  Clsdccld 7639

This theorem is referenced by:  iscld 7648

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-rex 1649  df-rab 1651  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fv 3195  df-cld 7642

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