Theorem fclusff 15623

Description: A function that takes a function from a class of functions to its cluster points.

Hypotheses

Ref	Expression
fclusff.1	|- X = U.J
fclusff.2	|- Y = U.L

Assertion

Ref	Expression
fclusff	|- ((J e. Top /\ L e. Fil) -> (J fClusf L) = {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))})

Distinct variable groups:   f,s,J   f,L,s   f,X,s   f,Y,s

Proof of Theorem fclusff

Step	Hyp	Ref	Expression
1		mapex 5387	. . . . . 6 |- ((Y e. _V /\ X e. _V) -> {g | g:Y-->X} e. _V)
2		uniexg 3795	. . . . . . 7 |- (L e. Fil -> U.L e. _V)
3		fclusff.2	. . . . . . 7 |- Y = U.L
4	2, 3	syl5eqel 1975	. . . . . 6 |- (L e. Fil -> Y e. _V)
5		uniexg 3795	. . . . . . 7 |- (J e. Top -> U.J e. _V)
6		fclusff.1	. . . . . . 7 |- X = U.J
7	5, 6	syl5eqel 1975	. . . . . 6 |- (J e. Top -> X e. _V)
8	1, 4, 7	syl2an 503	. . . . 5 |- ((L e. Fil /\ J e. Top) -> {g | g:Y-->X} e. _V)
9	8	ancoms 484	. . . 4 |- ((J e. Top /\ L e. Fil) -> {g | g:Y-->X} e. _V)
10		opabex2g 4540	. . . 4 |- ({g | g:Y-->X} e. _V -> {<.f, s>. | (f e. {g | g:Y-->X} /\ s = ((fClus` J)` ((X FilMap L)` f)))} e. _V)
11	9, 10	syl 12	. . 3 |- ((J e. Top /\ L e. Fil) -> {<.f, s>. | (f e. {g | g:Y-->X} /\ s = ((fClus` J)` ((X FilMap L)` f)))} e. _V)
12		visset 2295	. . . . . 6 |- f e. _V
13		feq1 4551	. . . . . 6 |- (g = f -> (g:Y-->X <-> f:Y-->X))
14	12, 13	elab 2403	. . . . 5 |- (f e. {g | g:Y-->X} <-> f:Y-->X)
15	14	anbi1i 539	. . . 4 |- ((f e. {g | g:Y-->X} /\ s = ((fClus` J)` ((X FilMap L)` f))) <-> (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f))))
16	15	opabbii 3402	. . 3 |- {<.f, s>. | (f e. {g | g:Y-->X} /\ s = ((fClus` J)` ((X FilMap L)` f)))} = {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))}
17	11, 16	syl5eqelr 1976	. 2 |- ((J e. Top /\ L e. Fil) -> {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))} e. _V)
18		unieq 3185	. . . . . . 7 |- (x = J -> U.x = U.J)
19	18, 6	syl6eqr 1946	. . . . . 6 |- (x = J -> U.x = X)
20		feq3 4553	. . . . . 6 |- (U.x = X -> (f:U.y-->U.x <-> f:U.y-->X))
21	19, 20	syl 12	. . . . 5 |- (x = J -> (f:U.y-->U.x <-> f:U.y-->X))
22		fveq2 4681	. . . . . . 7 |- (x = J -> (fClus` x) = (fClus` J))
23	19	opreq1d 4897	. . . . . . . 8 |- (x = J -> (U.x FilMap y) = (X FilMap y))
24	23	fveq1d 4683	. . . . . . 7 |- (x = J -> ((U.x FilMap y)` f) = ((X FilMap y)` f))
25	22, 24	fveq12d 10152	. . . . . 6 |- (x = J -> ((fClus` x)` ((U.x FilMap y)` f)) = ((fClus` J)` ((X FilMap y)` f)))
26	25	eqeq2d 1895	. . . . 5 |- (x = J -> (s = ((fClus` x)` ((U.x FilMap y)` f)) <-> s = ((fClus` J)` ((X FilMap y)` f))))
27	21, 26	anbi12d 690	. . . 4 |- (x = J -> ((f:U.y-->U.x /\ s = ((fClus` x)` ((U.x FilMap y)` f))) <-> (f:U.y-->X /\ s = ((fClus` J)` ((X FilMap y)` f)))))
28	27	opabbidv 3401	. . 3 |- (x = J -> {<.f, s>. | (f:U.y-->U.x /\ s = ((fClus` x)` ((U.x FilMap y)` f)))} = {<.f, s>. | (f:U.y-->X /\ s = ((fClus` J)` ((X FilMap y)` f)))})
29		unieq 3185	. . . . . . 7 |- (y = L -> U.y = U.L)
30	29, 3	syl6eqr 1946	. . . . . 6 |- (y = L -> U.y = Y)
31	30	feq2d 4557	. . . . 5 |- (y = L -> (f:U.y-->X <-> f:Y-->X))
32		opreq2 4890	. . . . . . . 8 |- (y = L -> (X FilMap y) = (X FilMap L))
33	32	fveq1d 4683	. . . . . . 7 |- (y = L -> ((X FilMap y)` f) = ((X FilMap L)` f))
34	33	fveq2d 4685	. . . . . 6 |- (y = L -> ((fClus` J)` ((X FilMap y)` f)) = ((fClus` J)` ((X FilMap L)` f)))
35	34	eqeq2d 1895	. . . . 5 |- (y = L -> (s = ((fClus` J)` ((X FilMap y)` f)) <-> s = ((fClus` J)` ((X FilMap L)` f))))
36	31, 35	anbi12d 690	. . . 4 |- (y = L -> ((f:U.y-->X /\ s = ((fClus` J)` ((X FilMap y)` f))) <-> (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))))
37	36	opabbidv 3401	. . 3 |- (y = L -> {<.f, s>. | (f:U.y-->X /\ s = ((fClus` J)` ((X FilMap y)` f)))} = {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))})
38		df-fclusf 15585	. . . 4 |- fClusf = {<.<.x, y>., z>. | (x e. Top /\ y e. Fil /\ z = {<.f, s>. | (f:U.y-->U.x /\ s = ((fClus` x)` ((U.x FilMap y)` f)))})}
39		df-3an 860	. . . . 5 |- ((x e. Top /\ y e. Fil /\ z = {<.f, s>. | (f:U.y-->U.x /\ s = ((fClus` x)` ((U.x FilMap y)` f)))}) <-> ((x e. Top /\ y e. Fil) /\ z = {<.f, s>. | (f:U.y-->U.x /\ s = ((fClus` x)` ((U.x FilMap y)` f)))}))
40	39	oprabbii 4923	. . . 4 |- {<.<.x, y>., z>. | (x e. Top /\ y e. Fil /\ z = {<.f, s>. | (f:U.y-->U.x /\ s = ((fClus` x)` ((U.x FilMap y)` f)))})} = {<.<.x, y>., z>. | ((x e. Top /\ y e. Fil) /\ z = {<.f, s>. | (f:U.y-->U.x /\ s = ((fClus` x)` ((U.x FilMap y)` f)))})}
41	38, 40	eqtri 1908	. . 3 |- fClusf = {<.<.x, y>., z>. | ((x e. Top /\ y e. Fil) /\ z = {<.f, s>. | (f:U.y-->U.x /\ s = ((fClus` x)` ((U.x FilMap y)` f)))})}
42	28, 37, 41	oprabval2g 4956	. 2 |- ((J e. Top /\ L e. Fil /\ {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))} e. _V) -> (J fClusf L) = {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))})
43	17, 42	mpd3an3 1192	1 |- ((J e. Top /\ L e. Fil) -> (J fClusf L) = {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292  U.cuni 3177  {copab 3395  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  Topctop 8857  Filcfil 10264   FilMap cfilmap 10304  fCluscfclus 15582   fClusf cfclusf 15583

This theorem is referenced by:  sfclusf 15624

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-3an 860  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-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  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-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-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-fclusf 15585

Copyright terms: Public domain