Theorem clsval 8953

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.

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

Step	Hyp	Ref	Expression
1		iscld.1	. . . . . 6 |- X = U.J
2	1	clsfval 8944	. . . . 5 |- (J e. Top -> (cls` J) = {<.y, z>. | (y C_ X /\ z = |^|{x e. (Clsd` J) | y C_ x})})
3	2	adantr 425	. . . 4 |- ((J e. Top /\ S C_ X) -> (cls` J) = {<.y, z>. | (y C_ X /\ z = |^|{x e. (Clsd` J) | y C_ x})})
4		visset 2295	. . . . . . 7 |- y e. _V
5	4	elpw 3037	. . . . . 6 |- (y e. ~PX <-> y C_ X)
6	5	anbi1i 539	. . . . 5 |- ((y e. ~PX /\ z = |^|{x e. (Clsd` J) | y C_ x}) <-> (y C_ X /\ z = |^|{x e. (Clsd` J) | y C_ x}))
7	6	opabbii 3402	. . . 4 |- {<.y, z>. | (y e. ~PX /\ z = |^|{x e. (Clsd` J) | y C_ x})} = {<.y, z>. | (y C_ X /\ z = |^|{x e. (Clsd` J) | y C_ x})}
8	3, 7	syl6eqr 1946	. . 3 |- ((J e. Top /\ S C_ X) -> (cls` J) = {<.y, z>. | (y e. ~PX /\ z = |^|{x e. (Clsd` J) | y C_ x})})
9	8	fveq1d 4683	. 2 |- ((J e. Top /\ S C_ X) -> ((cls` J)` S) = ({<.y, z>. | (y e. ~PX /\ z = |^|{x e. (Clsd` J) | y C_ x})}` S))
10		elpw2g 3463	. . . . 5 |- (X e. _V -> (S e. ~PX <-> S C_ X))
11	10	biimpar 461	. . . 4 |- ((X e. _V /\ S C_ X) -> S e. ~PX)
12		uniexg 3795	. . . . 5 |- (J e. Top -> U.J e. _V)
13	12, 1	syl5eqel 1975	. . . 4 |- (J e. Top -> X e. _V)
14	11, 13	sylan 497	. . 3 |- ((J e. Top /\ S C_ X) -> S e. ~PX)
15		sseq2 2639	. . . . . 6 |- (x = X -> (S C_ x <-> S C_ X))
16	15	rcla4ev 2381	. . . . 5 |- ((X e. (Clsd` J) /\ S C_ X) -> E.x e. (Clsd` J)S C_ x)
17	1	topcld 8951	. . . . 5 |- (J e. Top -> X e. (Clsd` J))
18	16, 17	sylan 497	. . . 4 |- ((J e. Top /\ S C_ X) -> E.x e. (Clsd` J)S C_ x)
19		intexrab 3468	. . . 4 |- (E.x e. (Clsd` J)S C_ x <-> |^|{x e. (Clsd` J) | S C_ x} e. _V)
20	18, 19	sylib 215	. . 3 |- ((J e. Top /\ S C_ X) -> |^|{x e. (Clsd` J) | S C_ x} e. _V)
21		sseq1 2637	. . . . . 6 |- (y = S -> (y C_ x <-> S C_ x))
22	21	rabbidv 2287	. . . . 5 |- (y = S -> {x e. (Clsd` J) | y C_ x} = {x e. (Clsd` J) | S C_ x})
23	22	inteqd 3219	. . . 4 |- (y = S -> |^|{x e. (Clsd` J) | y C_ x} = |^|{x e. (Clsd` J) | S C_ x})
24		eqid 1884	. . . 4 |- {<.y, z>. | (y e. ~PX /\ z = |^|{x e. (Clsd` J) | y C_ x})} = {<.y, z>. | (y e. ~PX /\ z = |^|{x e. (Clsd` J) | y C_ x})}
25	23, 24	fvopab4g 4742	. . 3 |- ((S e. ~PX /\ |^|{x e. (Clsd` J) | S C_ x} e. _V) -> ({<.y, z>. | (y e. ~PX /\ z = |^|{x e. (Clsd` J) | y C_ x})}` S) = |^|{x e. (Clsd` J) | S C_ x})
26	14, 20, 25	syl11anc 524	. 2 |- ((J e. Top /\ S C_ X) -> ({<.y, z>. | (y e. ~PX /\ z = |^|{x e. (Clsd` J) | y C_ x})}` S) = |^|{x e. (Clsd` J) | S C_ x})
27	9, 26	eqtrd 1925	1 |- ((J e. Top /\ S C_ X) -> ((cls` J)` S) = |^|{x e. (Clsd` J) | S C_ x})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032  U.cuni 3177  |^|cint 3214  {copab 3395  ` cfv 3998  Topctop 8857  Clsdccld 8936  clsccl 8938

This theorem is referenced by:  cldcls 8958  clscld 8959  clsval2 8961  clsss 8963  sscls 8965  islp2 9023

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-top 8861  df-cld 8939  df-cls 8941

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