Theorem cldval 15200

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 15109	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4317	. . 3 |- ( X e. J -> ~P X e. _V )
4		rabexg 4279	. . 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 3944	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2289	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3693	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9	7	difeq1d 3346	. . . . 5 |- ( j = J -> ( U. j \ x ) = ( X \ x ) )
10		eleq12 2303	. . . . 5 |- ( ( ( U. j \ x ) = ( X \ x ) /\ j = J ) -> ( ( U. j \ x ) e. j <-> ( X \ x ) e. J ) )
11	9, 10	mpancom 426	. . . 4 |- ( j = J -> ( ( U. j \ x ) e. j <-> ( X \ x ) e. J ) )
12	8, 11	rabeqbidv 2816	. . 3 |- ( j = J -> { x e. ~P U. j | ( U. j \ x ) e. j } = { x e. ~P X | ( X \ x ) e. J } )
13		df-cld 15196	. . 3 |- Clsd = ( j e. Top |-> { x e. ~P U. j | ( U. j \ x ) e. j } )
14	12, 13	fvmptg 5781	. 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 425	1 |- ( J e. Top -> ( Clsd ` J ) = { x e. ~P X | ( X \ x ) e. J } )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   \ cdif 3217   ~Pcpw 3688   U.cuni 3935   ` cfv 5377   Topctop 15098   Clsdccld 15193

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-top 15099  df-cld 15196

This theorem is used by:  iscld  15204