Theorem clsval 14431

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 14421	. . . 4 |- ( J e. Top -> ( cls ` J ) = ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) )
3	2	fveq1d 5563	. . 3 |- ( J e. Top -> ( ( cls ` J ) ` S ) = ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) )
4	3	adantr 276	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) )
5		eqid 2196	. . 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 3207	. . . . 5 |- ( y = S -> ( y C_ x <-> S C_ x ) )
7	6	rabbidv 2752	. . . 4 |- ( y = S -> { x e. ( Clsd ` J ) | y C_ x } = { x e. ( Clsd ` J ) | S C_ x } )
8	7	inteqd 3880	. . 3 |- ( y = S -> |^| { x e. ( Clsd ` J ) | y C_ x } = |^| { x e. ( Clsd ` J ) | S C_ x } )
9	1	topopn 14328	. . . . 5 |- ( J e. Top -> X e. J )
10		elpw2g 4190	. . . . 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 297	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
13	1	topcld 14429	. . . . 5 |- ( J e. Top -> X e. ( Clsd ` J ) )
14		sseq2 3208	. . . . . 6 |- ( x = X -> ( S C_ x <-> S C_ X ) )
15	14	rspcev 2868	. . . . 5 |- ( ( X e. ( Clsd ` J ) /\ S C_ X ) -> E. x e. ( Clsd ` J ) S C_ x )
16	13, 15	sylan 283	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> E. x e. ( Clsd ` J ) S C_ x )
17		intexrabim 4187	. . . 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 5658	. 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 2229	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   E.wrex 2476   {crab 2479   _Vcvv 2763   C_ wss 3157   ~Pcpw 3606   U.cuni 3840   |^|cint 3875   |-> cmpt 4095   ` cfv 5259   Topctop 14317   Clsdccld 14412   clsccl 14414

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-top 14318  df-cld 14415  df-cls 14417

This theorem is referenced by:  cldcls  14434  clsss  14438  sscls  14440