Theorem cldval 14571

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 14480	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4224	. . 3 |- ( X e. J -> ~P X e. _V )
4		rabexg 4187	. . 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 3859	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2256	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3621	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9	7	difeq1d 3290	. . . . 5 |- ( j = J -> ( U. j \ x ) = ( X \ x ) )
10		eleq12 2270	. . . . 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 2767	. . 3 |- ( j = J -> { x e. ~P U. j | ( U. j \ x ) e. j } = { x e. ~P X | ( X \ x ) e. J } )
13		df-cld 14567	. . 3 |- Clsd = ( j e. Top |-> { x e. ~P U. j | ( U. j \ x ) e. j } )
14	12, 13	fvmptg 5655	. 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 1373   e. wcel 2176   {crab 2488   _Vcvv 2772   \ cdif 3163   ~Pcpw 3616   U.cuni 3850   ` cfv 5271   Topctop 14469   Clsdccld 14564

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-top 14470  df-cld 14567

This theorem is referenced by:  iscld  14575