Theorem lpfval 7742

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

Hypothesis

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

Assertion

Ref	Expression
lpfval	|- (J e. Top -> (limPt` J) = {<.x, y>. | (x (_ X /\ y = {v | v e. ((cls` J)` (x \ {v}))})})

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

Proof of Theorem lpfval

Step	Hyp	Ref	Expression
1		uniexg 2871	. . . . 5 |- (J e. Top -> U.J e. V)
2		lpfval.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 | v e. ((cls` J)` (x \ {v}))})} e. V)
6	3, 4, 5	3syl 20	. . 3 |- (J e. Top -> {<.x, y>. | (x e. P~X /\ y = {v | v e. ((cls` 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 | v e. ((cls` J)` (x \ {v}))}) <-> (x (_ X /\ y = {v | v e. ((cls` J)` (x \ {v}))}))
10	9	opabbii 2671	. . 3 |- {<.x, y>. | (x e. P~X /\ y = {v | v e. ((cls` J)` (x \ {v}))})} = {<.x, y>. | (x (_ X /\ y = {v | v e. ((cls` J)` (x \ {v}))})}
11	6, 10	syl5eqelr 1553	. 2 |- (J e. Top -> {<.x, y>. | (x (_ X /\ y = {v | v e. ((cls` 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	. . . . . . . . 9 |- (z = J -> (cls` z) = (cls` J))
16	15	fveq1d 3726	. . . . . . . 8 |- (z = J -> ((cls` z)` (x \ {v})) = ((cls` J)` (x \ {v})))
17	16	eleq2d 1541	. . . . . . 7 |- (z = J -> (v e. ((cls` z)` (x \ {v})) <-> v e. ((cls` J)` (x \ {v}))))
18	17	abbidv 1577	. . . . . 6 |- (z = J -> {v | v e. ((cls` z)` (x \ {v}))} = {v | v e. ((cls` J)` (x \ {v}))})
19	18	eqeq2d 1486	. . . . 5 |- (z = J -> (y = {v | v e. ((cls` z)` (x \ {v}))} <-> y = {v | v e. ((cls` J)` (x \ {v}))}))
20	14, 19	anbi12d 628	. . . 4 |- (z = J -> ((x (_ U.z /\ y = {v | v e. ((cls` z)` (x \ {v}))}) <-> (x (_ X /\ y = {v | v e. ((cls` J)` (x \ {v}))})))
21	20	opabbidv 2670	. . 3 |- (z = J -> {<.x, y>. | (x (_ U.z /\ y = {v | v e. ((cls` z)` (x \ {v}))})} = {<.x, y>. | (x (_ X /\ y = {v | v e. ((cls` J)` (x \ {v}))})})
22		df-lp 7741	. . 3 |- limPt = {<.z, w>. | (z e. Top /\ w = {<.x, y>. | (x (_ U.z /\ y = {v | v e. ((cls` z)` (x \ {v}))})})}
23	21, 22	fvopab4g 3779	. 2 |- ((J e. Top /\ {<.x, y>. | (x (_ X /\ y = {v | v e. ((cls` J)` (x \ {v}))})} e. V) -> (limPt` J) = {<.x, y>. | (x (_ X /\ y = {v | v e. ((cls` J)` (x \ {v}))})})
24	11, 23	mpdan 704	1 |- (J e. Top -> (limPt` J) = {<.x, y>. | (x (_ X /\ y = {v | v e. ((cls` J)` (x \ {v}))})})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   \ cdif 2044   (_ wss 2047  P~cpw 2401  {csn 2409  U.cuni 2503  {copab 2666  ` cfv 3182  Topctop 7588  clsccl 7662  limPtclp 7740

This theorem is referenced by:  lpval 7743

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-rex 1650  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-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-lp 7741

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