Theorem clscld 7683

Description: The closure of a subset of a topology's underlying set is closed.

Hypothesis

Ref	Expression
clscld.1	|- X = U.J

Assertion

Ref	Expression
clscld	|- ((J e. Top /\ S (_ X) -> ((cls` J)` S) e. (Clsd` J))

Proof of Theorem clscld

Step	Hyp	Ref	Expression
1		clscld.1	. . 3 |- X = U.J
2	1	clsval 7677	. 2 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) = |^|{x e. (Clsd` J) | S (_ x})
3		intcld 7680	. . 3 |- ((J e. Top /\ {x e. (Clsd` J) | S (_ x} =/= (/) /\ {x e. (Clsd` J) | S (_ x} (_ (Clsd` J)) -> |^|{x e. (Clsd` J) | S (_ x} e. (Clsd` J))
4		pm3.26 319	. . 3 |- ((J e. Top /\ S (_ X) -> J e. Top)
5	1	topcld 7675	. . . . . 6 |- (J e. Top -> X e. (Clsd` J))
6	5	anim1i 334	. . . . 5 |- ((J e. Top /\ S (_ X) -> (X e. (Clsd` J) /\ S (_ X))
7		sseq2 2083	. . . . . 6 |- (x = X -> (S (_ x <-> S (_ X))
8	7	elrab 1905	. . . . 5 |- (X e. {x e. (Clsd` J) | S (_ x} <-> (X e. (Clsd` J) /\ S (_ X))
9	6, 8	sylibr 200	. . . 4 |- ((J e. Top /\ S (_ X) -> X e. {x e. (Clsd` J) | S (_ x})
10		ne0i 2286	. . . 4 |- (X e. {x e. (Clsd` J) | S (_ x} -> {x e. (Clsd` J) | S (_ x} =/= (/))
11	9, 10	syl 10	. . 3 |- ((J e. Top /\ S (_ X) -> {x e. (Clsd` J) | S (_ x} =/= (/))
12		ssrab2 2131	. . . 4 |- {x e. (Clsd` J) | S (_ x} (_ (Clsd` J)
13	12	a1i 8	. . 3 |- ((J e. Top /\ S (_ X) -> {x e. (Clsd` J) | S (_ x} (_ (Clsd` J))
14	3, 4, 11, 13	syl3anc 858	. 2 |- ((J e. Top /\ S (_ X) -> |^|{x e. (Clsd` J) | S (_ x} e. (Clsd` J))
15	2, 14	eqeltrd 1548	1 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) e. (Clsd` J))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585  {crab 1648   (_ wss 2047  (/)c0 2280  U.cuni 2503  |^|cint 2533  ` cfv 3182  Topctop 7588  Clsdccld 7660  clsccl 7662

This theorem is referenced by:  clsss3 7691  cmntrcld 7694  iscld3 7695  clsidm 7698

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-3an 777  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-sbc 1942  df-csb 2002  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-iun 2568  df-iin 2569  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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