Theorem clsval 7619

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 (_ X) -> ((cls` J)` S) = |^|{x e. (Clsd` J) | S (_ 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 7610	. . . . 5 |- (J e. Top -> (cls` J) = {<.y, z>. | (y (_ X /\ z = |^|{x e. (Clsd` J) | y (_ x})})
3	2	adantr 389	. . . 4 |- ((J e. Top /\ S (_ X) -> (cls` J) = {<.y, z>. | (y (_ X /\ z = |^|{x e. (Clsd` J) | y (_ x})})
4		visset 1804	. . . . . . 7 |- y e. V
5	4	elpw 2394	. . . . . 6 |- (y e. P~X <-> y (_ X)
6	5	anbi1i 480	. . . . 5 |- ((y e. P~X /\ z = |^|{x e. (Clsd` J) | y (_ x}) <-> (y (_ X /\ z = |^|{x e. (Clsd` J) | y (_ x}))
7	6	opabbii 2661	. . . 4 |- {<.y, z>. | (y e. P~X /\ z = |^|{x e. (Clsd` J) | y (_ x})} = {<.y, z>. | (y (_ X /\ z = |^|{x e. (Clsd` J) | y (_ x})}
8	3, 7	syl6eqr 1517	. . 3 |- ((J e. Top /\ S (_ X) -> (cls` J) = {<.y, z>. | (y e. P~X /\ z = |^|{x e. (Clsd` J) | y (_ x})})
9	8	fveq1d 3711	. 2 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) = ({<.y, z>. | (y e. P~X /\ z = |^|{x e. (Clsd` J) | y (_ x})}` S))
10		sseq1 2072	. . . . . 6 |- (y = S -> (y (_ x <-> S (_ x))
11	10	rabbisdv 1798	. . . . 5 |- (y = S -> {x e. (Clsd` J) | y (_ x} = {x e. (Clsd` J) | S (_ x})
12	11	inteqd 2528	. . . 4 |- (y = S -> |^|{x e. (Clsd` J) | y (_ x} = |^|{x e. (Clsd` J) | S (_ x})
13		eqid 1468	. . . 4 |- {<.y, z>. | (y e. P~X /\ z = |^|{x e. (Clsd` J) | y (_ x})} = {<.y, z>. | (y e. P~X /\ z = |^|{x e. (Clsd` J) | y (_ x})}
14	12, 13	fvopab4g 3764	. . 3 |- ((S e. P~X /\ |^|{x e. (Clsd` J) | S (_ x} e. V) -> ({<.y, z>. | (y e. P~X /\ z = |^|{x e. (Clsd` J) | y (_ x})}` S) = |^|{x e. (Clsd` J) | S (_ x})
15		elpw2g 2717	. . . . 5 |- (X e. V -> (S e. P~X <-> S (_ X))
16	15	biimpar 417	. . . 4 |- ((X e. V /\ S (_ X) -> S e. P~X)
17		uniexg 2862	. . . . 5 |- (J e. Top -> U.J e. V)
18	17, 1	syl5eqel 1544	. . . 4 |- (J e. Top -> X e. V)
19	16, 18	sylan 448	. . 3 |- ((J e. Top /\ S (_ X) -> S e. P~X)
20		sseq2 2073	. . . . . 6 |- (x = X -> (S (_ x <-> S (_ X))
21	20	rcla4ev 1868	. . . . 5 |- ((X e. (Clsd` J) /\ S (_ X) -> E.x e. (Clsd` J)S (_ x)
22	1	topcld 7617	. . . . 5 |- (J e. Top -> X e. (Clsd` J))
23	21, 22	sylan 448	. . . 4 |- ((J e. Top /\ S (_ X) -> E.x e. (Clsd` J)S (_ x)
24		intexrab 2722	. . . 4 |- (E.x e. (Clsd` J)S (_ x <-> |^|{x e. (Clsd` J) | S (_ x} e. V)
25	23, 24	sylib 198	. . 3 |- ((J e. Top /\ S (_ X) -> |^|{x e. (Clsd` J) | S (_ x} e. V)
26	14, 19, 25	sylanc 471	. 2 |- ((J e. Top /\ S (_ X) -> ({<.y, z>. | (y e. P~X /\ z = |^|{x e. (Clsd` J) | y (_ x})}` S) = |^|{x e. (Clsd` J) | S (_ x})
27	9, 26	eqtrd 1499	1 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) = |^|{x e. (Clsd` J) | S (_ x})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wrex 1638  {crab 1640  Vcvv 1802   (_ wss 2037  P~cpw 2391  U.cuni 2493  |^|cint 2523  {copab 2656  ` cfv 3172  Topctop 7530  Clsdccld 7602  clsccl 7604

This theorem is referenced by:  cldcls 7624  clscld 7625  clsval2 7627  clsss 7629  sscls 7631  islp2 7688

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-top 7534  df-cld 7605  df-cls 7607

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