Theorem clsval 7677

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 7668	. . . . 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 1813	. . . . . . 7 |- y e. V
5	4	elpw 2404	. . . . . 6 |- (y e. P~X <-> y (_ X)
6	5	anbi1i 481	. . . . 5 |- ((y e. P~X /\ z = |^|{x e. (Clsd` J) | y (_ x}) <-> (y (_ X /\ z = |^|{x e. (Clsd` J) | y (_ x}))
7	6	opabbii 2671	. . . 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 1525	. . 3 |- ((J e. Top /\ S (_ X) -> (cls` J) = {<.y, z>. | (y e. P~X /\ z = |^|{x e. (Clsd` J) | y (_ x})})
9	8	fveq1d 3726	. 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 2082	. . . . . 6 |- (y = S -> (y (_ x <-> S (_ x))
11	10	rabbisdv 1807	. . . . 5 |- (y = S -> {x e. (Clsd` J) | y (_ x} = {x e. (Clsd` J) | S (_ x})
12	11	inteqd 2538	. . . 4 |- (y = S -> |^|{x e. (Clsd` J) | y (_ x} = |^|{x e. (Clsd` J) | S (_ x})
13		eqid 1475	. . . 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 3779	. . 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 2727	. . . . 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 2871	. . . . 5 |- (J e. Top -> U.J e. V)
18	17, 1	syl5eqel 1552	. . . 4 |- (J e. Top -> X e. V)
19	16, 18	sylan 448	. . 3 |- ((J e. Top /\ S (_ X) -> S e. P~X)
20		sseq2 2083	. . . . . 6 |- (x = X -> (S (_ x <-> S (_ X))
21	20	rcla4ev 1877	. . . . 5 |- ((X e. (Clsd` J) /\ S (_ X) -> E.x e. (Clsd` J)S (_ x)
22	1	topcld 7675	. . . . 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 2732	. . . 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 1507	1 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) = |^|{x e. (Clsd` J) | S (_ x})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wrex 1646  {crab 1648  Vcvv 1811   (_ wss 2047  P~cpw 2401  U.cuni 2503  |^|cint 2533  {copab 2666  ` cfv 3182  Topctop 7588  Clsdccld 7660  clsccl 7662

This theorem is referenced by:  cldcls 7682  clscld 7683  clsval2 7685  clsss 7687  sscls 7689  islp2 7747

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-top 7592  df-cld 7663  df-cls 7665

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