Theorem cldval 12739

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 12646	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4159	. . 3 |- ( X e. J -> ~P X e. _V )
4		rabexg 4125	. . 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 3798	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2217	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3564	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9	7	difeq1d 3239	. . . . 5 |- ( j = J -> ( U. j \ x ) = ( X \ x ) )
10		eleq12 2231	. . . . 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 2721	. . 3 |- ( j = J -> { x e. ~P U. j | ( U. j \ x ) e. j } = { x e. ~P X | ( X \ x ) e. J } )
13		df-cld 12735	. . 3 |- Clsd = ( j e. Top |-> { x e. ~P U. j | ( U. j \ x ) e. j } )
14	12, 13	fvmptg 5562	. 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 1343   e. wcel 2136   {crab 2448   _Vcvv 2726   \ cdif 3113   ~Pcpw 3559   U.cuni 3789   ` cfv 5188   Topctop 12635   Clsdccld 12732

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-top 12636  df-cld 12735

This theorem is referenced by:  iscld  12743