Theorem clsval 20101

Description: The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

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

Assertion

Ref	Expression
clsval	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )

Distinct variable groups:   x, J   x, S   x, X

Proof of Theorem clsval

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscld.1	. . . . 5 |- X = U. J
2	1	clsfval 20089	. . . 4 |- ( J e. Top -> ( cls ` J ) = ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) )
3	2	fveq1d 5890	. . 3 |- ( J e. Top -> ( ( cls ` J ) ` S ) = ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) )
4	3	adantr 471	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) )
5	1	topopn 19985	. . . . 5 |- ( J e. Top -> X e. J )
6		elpw2g 4580	. . . . 5 |- ( X e. J -> ( S e. ~P X <-> S C_ X ) )
7	5, 6	syl 17	. . . 4 |- ( J e. Top -> ( S e. ~P X <-> S C_ X ) )
8	7	biimpar 492	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
9	1	topcld 20099	. . . . 5 |- ( J e. Top -> X e. ( Clsd ` J ) )
10		sseq2 3466	. . . . . 6 |- ( x = X -> ( S C_ x <-> S C_ X ) )
11	10	rspcev 3162	. . . . 5 |- ( ( X e. ( Clsd ` J ) /\ S C_ X ) -> E. x e. ( Clsd ` J ) S C_ x )
12	9, 11	sylan 478	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> E. x e. ( Clsd ` J ) S C_ x )
13		intexrab 4576	. . . 4 |- ( E. x e. ( Clsd ` J ) S C_ x <-> |^| { x e. ( Clsd ` J ) | S C_ x } e. _V )
14	12, 13	sylib 201	. . 3 |- ( ( J e. Top /\ S C_ X ) -> |^| { x e. ( Clsd ` J ) | S C_ x } e. _V )
15		sseq1 3465	. . . . . 6 |- ( y = S -> ( y C_ x <-> S C_ x ) )
16	15	rabbidv 3048	. . . . 5 |- ( y = S -> { x e. ( Clsd ` J ) | y C_ x } = { x e. ( Clsd ` J ) | S C_ x } )
17	16	inteqd 4253	. . . 4 |- ( y = S -> |^| { x e. ( Clsd ` J ) | y C_ x } = |^| { x e. ( Clsd ` J ) | S C_ x } )
18		eqid 2462	. . . 4 |- ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) = ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } )
19	17, 18	fvmptg 5969	. . 3 |- ( ( S e. ~P X /\ |^| { x e. ( Clsd ` J ) | S C_ x } e. _V ) -> ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )
20	8, 14, 19	syl2anc 671	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )
21	4, 20	eqtrd 2496	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   E.wrex 2750   {crab 2753   _Vcvv 3057   C_ wss 3416   ~Pcpw 3963   U.cuni 4212   |^|cint 4248   |-> cmpt 4475   ` cfv 5601   Topctop 19966   Clsdccld 20080   clsccl 20082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-top 19970  df-cld 20083  df-cls 20085

This theorem is referenced by:  cldcls  20106  clscld  20111  clsf  20112  clsval2  20114  clsss  20118  sscls  20120