Theorem clsval 12751

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 12741	. . . 4 |- ( J e. Top -> ( cls ` J ) = ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) )
3	2	fveq1d 5488	. . 3 |- ( J e. Top -> ( ( cls ` J ) ` S ) = ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) )
4	3	adantr 274	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) )
5		eqid 2165	. . 3 |- ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) = ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } )
6		sseq1 3165	. . . . 5 |- ( y = S -> ( y C_ x <-> S C_ x ) )
7	6	rabbidv 2715	. . . 4 |- ( y = S -> { x e. ( Clsd ` J ) | y C_ x } = { x e. ( Clsd ` J ) | S C_ x } )
8	7	inteqd 3829	. . 3 |- ( y = S -> |^| { x e. ( Clsd ` J ) | y C_ x } = |^| { x e. ( Clsd ` J ) | S C_ x } )
9	1	topopn 12646	. . . . 5 |- ( J e. Top -> X e. J )
10		elpw2g 4135	. . . . 5 |- ( X e. J -> ( S e. ~P X <-> S C_ X ) )
11	9, 10	syl 14	. . . 4 |- ( J e. Top -> ( S e. ~P X <-> S C_ X ) )
12	11	biimpar 295	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
13	1	topcld 12749	. . . . 5 |- ( J e. Top -> X e. ( Clsd ` J ) )
14		sseq2 3166	. . . . . 6 |- ( x = X -> ( S C_ x <-> S C_ X ) )
15	14	rspcev 2830	. . . . 5 |- ( ( X e. ( Clsd ` J ) /\ S C_ X ) -> E. x e. ( Clsd ` J ) S C_ x )
16	13, 15	sylan 281	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> E. x e. ( Clsd ` J ) S C_ x )
17		intexrabim 4132	. . . 4 |- ( E. x e. ( Clsd ` J ) S C_ x -> |^| { x e. ( Clsd ` J ) | S C_ x } e. _V )
18	16, 17	syl 14	. . 3 |- ( ( J e. Top /\ S C_ X ) -> |^| { x e. ( Clsd ` J ) | S C_ x } e. _V )
19	5, 8, 12, 18	fvmptd3 5579	. 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 } )
20	4, 19	eqtrd 2198	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   {crab 2448   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559   U.cuni 3789   |^|cint 3824   |-> cmpt 4043   ` cfv 5188   Topctop 12635   Clsdccld 12732   clsccl 12734

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-in1 604  ax-in2 605  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-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  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-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  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-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-top 12636  df-cld 12735  df-cls 12737

This theorem is referenced by:  cldcls  12754  clsss  12758  sscls  12760