Theorem clsfval 8944

Description: The closure function on the subsets of a topology's base set.

Hypothesis

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

Assertion

Ref	Expression
clsfval	|- (J e. Top -> (cls` J) = {<.x, y>. | (x C_ X /\ y = |^|{v e. (Clsd` J) | x C_ v})})

Distinct variable groups:   x,v,y,J   x,X,y

Proof of Theorem clsfval

Step	Hyp	Ref	Expression
1		uniexg 3795	. . . . 5 |- (J e. Top -> U.J e. _V)
2		cldval.1	. . . . 5 |- X = U.J
3	1, 2	syl5eqel 1975	. . . 4 |- (J e. Top -> X e. _V)
4		pwexg 3489	. . . 4 |- (X e. _V -> ~PX e. _V)
5		opabex2g 4540	. . . 4 |- (~PX e. _V -> {<.x, y>. | (x e. ~PX /\ y = |^|{v e. (Clsd` J) | x C_ v})} e. _V)
6	3, 4, 5	3syl 24	. . 3 |- (J e. Top -> {<.x, y>. | (x e. ~PX /\ y = |^|{v e. (Clsd` J) | x C_ v})} e. _V)
7		visset 2295	. . . . . 6 |- x e. _V
8	7	elpw 3037	. . . . 5 |- (x e. ~PX <-> x C_ X)
9	8	anbi1i 539	. . . 4 |- ((x e. ~PX /\ y = |^|{v e. (Clsd` J) | x C_ v}) <-> (x C_ X /\ y = |^|{v e. (Clsd` J) | x C_ v}))
10	9	opabbii 3402	. . 3 |- {<.x, y>. | (x e. ~PX /\ y = |^|{v e. (Clsd` J) | x C_ v})} = {<.x, y>. | (x C_ X /\ y = |^|{v e. (Clsd` J) | x C_ v})}
11	6, 10	syl5eqelr 1976	. 2 |- (J e. Top -> {<.x, y>. | (x C_ X /\ y = |^|{v e. (Clsd` J) | x C_ v})} e. _V)
12		unieq 3185	. . . . . . 7 |- (z = J -> U.z = U.J)
13	12, 2	syl6eqr 1946	. . . . . 6 |- (z = J -> U.z = X)
14	13	sseq2d 2645	. . . . 5 |- (z = J -> (x C_ U.z <-> x C_ X))
15		fveq2 4681	. . . . . . . 8 |- (z = J -> (Clsd` z) = (Clsd` J))
16		rabeq 2289	. . . . . . . 8 |- ((Clsd` z) = (Clsd` J) -> {v e. (Clsd` z) | x C_ v} = {v e. (Clsd` J) | x C_ v})
17	15, 16	syl 12	. . . . . . 7 |- (z = J -> {v e. (Clsd` z) | x C_ v} = {v e. (Clsd` J) | x C_ v})
18	17	inteqd 3219	. . . . . 6 |- (z = J -> |^|{v e. (Clsd` z) | x C_ v} = |^|{v e. (Clsd` J) | x C_ v})
19	18	eqeq2d 1895	. . . . 5 |- (z = J -> (y = |^|{v e. (Clsd` z) | x C_ v} <-> y = |^|{v e. (Clsd` J) | x C_ v}))
20	14, 19	anbi12d 690	. . . 4 |- (z = J -> ((x C_ U.z /\ y = |^|{v e. (Clsd` z) | x C_ v}) <-> (x C_ X /\ y = |^|{v e. (Clsd` J) | x C_ v})))
21	20	opabbidv 3401	. . 3 |- (z = J -> {<.x, y>. | (x C_ U.z /\ y = |^|{v e. (Clsd` z) | x C_ v})} = {<.x, y>. | (x C_ X /\ y = |^|{v e. (Clsd` J) | x C_ v})})
22		df-cls 8941	. . 3 |- cls = {<.z, w>. | (z e. Top /\ w = {<.x, y>. | (x C_ U.z /\ y = |^|{v e. (Clsd` z) | x C_ v})})}
23	21, 22	fvopab4g 4742	. 2 |- ((J e. Top /\ {<.x, y>. | (x C_ X /\ y = |^|{v e. (Clsd` J) | x C_ v})} e. _V) -> (cls` J) = {<.x, y>. | (x C_ X /\ y = |^|{v e. (Clsd` J) | x C_ v})})
24	11, 23	mpdan 768	1 |- (J e. Top -> (cls` J) = {<.x, y>. | (x C_ X /\ y = |^|{v e. (Clsd` J) | x C_ v})})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {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:  clsval 8953

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-cls 8941

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