Theorem clsfval 7668

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 (_ X /\ y = |^|{v e. (Clsd` J) | x (_ v})})

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

Proof of Theorem clsfval

Step	Hyp	Ref	Expression
1		uniexg 2871	. . . . 5 |- (J e. Top -> U.J e. V)
2		cldval.1	. . . . 5 |- X = U.J
3	1, 2	syl5eqel 1552	. . . 4 |- (J e. Top -> X e. V)
4		pwexg 2746	. . . 4 |- (X e. V -> P~X e. V)
5		opabex2g 3611	. . . 4 |- (P~X e. V -> {<.x, y>. | (x e. P~X /\ y = |^|{v e. (Clsd` J) | x (_ v})} e. V)
6	3, 4, 5	3syl 20	. . 3 |- (J e. Top -> {<.x, y>. | (x e. P~X /\ y = |^|{v e. (Clsd` J) | x (_ v})} e. V)
7		visset 1813	. . . . . 6 |- x e. V
8	7	elpw 2404	. . . . 5 |- (x e. P~X <-> x (_ X)
9	8	anbi1i 481	. . . 4 |- ((x e. P~X /\ y = |^|{v e. (Clsd` J) | x (_ v}) <-> (x (_ X /\ y = |^|{v e. (Clsd` J) | x (_ v}))
10	9	opabbii 2671	. . 3 |- {<.x, y>. | (x e. P~X /\ y = |^|{v e. (Clsd` J) | x (_ v})} = {<.x, y>. | (x (_ X /\ y = |^|{v e. (Clsd` J) | x (_ v})}
11	6, 10	syl5eqelr 1553	. 2 |- (J e. Top -> {<.x, y>. | (x (_ X /\ y = |^|{v e. (Clsd` J) | x (_ v})} e. V)
12		unieq 2510	. . . . . . 7 |- (z = J -> U.z = U.J)
13	12, 2	syl6eqr 1525	. . . . . 6 |- (z = J -> U.z = X)
14	13	sseq2d 2089	. . . . 5 |- (z = J -> (x (_ U.z <-> x (_ X))
15		fveq2 3724	. . . . . . . 8 |- (z = J -> (Clsd` z) = (Clsd` J))
16		rabeq 1809	. . . . . . . 8 |- ((Clsd` z) = (Clsd` J) -> {v e. (Clsd` z) | x (_ v} = {v e. (Clsd` J) | x (_ v})
17	15, 16	syl 10	. . . . . . 7 |- (z = J -> {v e. (Clsd` z) | x (_ v} = {v e. (Clsd` J) | x (_ v})
18	17	inteqd 2538	. . . . . 6 |- (z = J -> |^|{v e. (Clsd` z) | x (_ v} = |^|{v e. (Clsd` J) | x (_ v})
19	18	eqeq2d 1486	. . . . 5 |- (z = J -> (y = |^|{v e. (Clsd` z) | x (_ v} <-> y = |^|{v e. (Clsd` J) | x (_ v}))
20	14, 19	anbi12d 628	. . . 4 |- (z = J -> ((x (_ U.z /\ y = |^|{v e. (Clsd` z) | x (_ v}) <-> (x (_ X /\ y = |^|{v e. (Clsd` J) | x (_ v})))
21	20	opabbidv 2670	. . 3 |- (z = J -> {<.x, y>. | (x (_ U.z /\ y = |^|{v e. (Clsd` z) | x (_ v})} = {<.x, y>. | (x (_ X /\ y = |^|{v e. (Clsd` J) | x (_ v})})
22		df-cls 7665	. . 3 |- cls = {<.z, w>. | (z e. Top /\ w = {<.x, y>. | (x (_ U.z /\ y = |^|{v e. (Clsd` z) | x (_ v})})}
23	21, 22	fvopab4g 3779	. 2 |- ((J e. Top /\ {<.x, y>. | (x (_ X /\ y = |^|{v e. (Clsd` J) | x (_ v})} e. V) -> (cls` J) = {<.x, y>. | (x (_ X /\ y = |^|{v e. (Clsd` J) | x (_ v})})
24	11, 23	mpdan 704	1 |- (J e. Top -> (cls` J) = {<.x, y>. | (x (_ X /\ y = |^|{v e. (Clsd` J) | x (_ v})})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {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:  clsval 7677

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-cls 7665

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