Theorem cldval 12307

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.) (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

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

Assertion

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

Distinct variable groups:   x, J   x, X

Proof of Theorem cldval

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cldval.1	. . . 4 |- X = U. J
2	1	topopn 12214	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4112	. . 3 |- ( X e. J -> ~P X e. _V )
4		rabexg 4079	. . 3 |- ( ~P X e. _V -> { x e. ~P X | ( X \ x ) e. J } e. _V )
5	2, 3, 4	3syl 17	. 2 |- ( J e. Top -> { x e. ~P X | ( X \ x ) e. J } e. _V )
6		unieq 3753	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2191	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3520	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9	7	difeq1d 3198	. . . . 5 |- ( j = J -> ( U. j \ x ) = ( X \ x ) )
10		eleq12 2205	. . . . 5 |- ( ( ( U. j \ x ) = ( X \ x ) /\ j = J ) -> ( ( U. j \ x ) e. j <-> ( X \ x ) e. J ) )
11	9, 10	mpancom 419	. . . 4 |- ( j = J -> ( ( U. j \ x ) e. j <-> ( X \ x ) e. J ) )
12	8, 11	rabeqbidv 2684	. . 3 |- ( j = J -> { x e. ~P U. j | ( U. j \ x ) e. j } = { x e. ~P X | ( X \ x ) e. J } )
13		df-cld 12303	. . 3 |- Clsd = ( j e. Top |-> { x e. ~P U. j | ( U. j \ x ) e. j } )
14	12, 13	fvmptg 5505	. 2 |- ( ( J e. Top /\ { x e. ~P X | ( X \ x ) e. J } e. _V ) -> ( Clsd ` J ) = { x e. ~P X | ( X \ x ) e. J } )
15	5, 14	mpdan 418	1 |- ( J e. Top -> ( Clsd ` J ) = { x e. ~P X | ( X \ x ) e. J } )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   e. wcel 1481   {crab 2421   _Vcvv 2689   \ cdif 3073   ~Pcpw 3515   U.cuni 3744   ` cfv 5131   Topctop 12203   Clsdccld 12300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-top 12204  df-cld 12303

This theorem is referenced by:  iscld  12311