Theorem cldval 13684

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 13593	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4182	. . 3 |- ( X e. J -> ~P X e. _V )
4		rabexg 4148	. . 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 3820	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2228	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3582	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9	7	difeq1d 3254	. . . . 5 |- ( j = J -> ( U. j \ x ) = ( X \ x ) )
10		eleq12 2242	. . . . 5 |- ( ( ( U. j \ x ) = ( X \ x ) /\ j = J ) -> ( ( U. j \ x ) e. j <-> ( X \ x ) e. J ) )
11	9, 10	mpancom 422	. . . 4 |- ( j = J -> ( ( U. j \ x ) e. j <-> ( X \ x ) e. J ) )
12	8, 11	rabeqbidv 2734	. . 3 |- ( j = J -> { x e. ~P U. j | ( U. j \ x ) e. j } = { x e. ~P X | ( X \ x ) e. J } )
13		df-cld 13680	. . 3 |- Clsd = ( j e. Top |-> { x e. ~P U. j | ( U. j \ x ) e. j } )
14	12, 13	fvmptg 5594	. 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 421	1 |- ( J e. Top -> ( Clsd ` J ) = { x e. ~P X | ( X \ x ) e. J } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   {crab 2459   _Vcvv 2739   \ cdif 3128   ~Pcpw 3577   U.cuni 3811   ` cfv 5218   Topctop 13582   Clsdccld 13677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-top 13583  df-cld 13680

This theorem is referenced by:  iscld  13688