Theorem sfclusf 15624

Description: The set of cluster points of a function.

Hypotheses

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

Assertion

Ref	Expression
sfclusf	|- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> ((J fClusf L)` F) = ((fClus` J)` ((X FilMap L)` F)))

Proof of Theorem sfclusf

Step	Hyp	Ref	Expression
1		simpr 350	. . . . 5 |- ((f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f))) -> s = ((fClus` J)` ((X FilMap L)` f)))
2	1	ssopab2i 3574	. . . 4 |- {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))} C_ {<.f, s>. | s = ((fClus` J)` ((X FilMap L)` f))}
3		funopabeq 4456	. . . 4 |- Fun {<.f, s>. | s = ((fClus` J)` ((X FilMap L)` f))}
4		funss 4439	. . . 4 |- ({<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))} C_ {<.f, s>. | s = ((fClus` J)` ((X FilMap L)` f))} -> (Fun {<.f, s>. | s = ((fClus` J)` ((X FilMap L)` f))} -> Fun {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))}))
5	2, 3, 4	mp2 54	. . 3 |- Fun {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))}
6		sfclusf.1	. . . . . 6 |- X = U.J
7		sfclusf.2	. . . . . 6 |- Y = U.L
8	6, 7	fclusff 15623	. . . . 5 |- ((J e. Top /\ L e. Fil) -> (J fClusf L) = {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))})
9		funeq 4441	. . . . 5 |- ((J fClusf L) = {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))} -> (Fun (J fClusf L) <-> Fun {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))}))
10	8, 9	syl 12	. . . 4 |- ((J e. Top /\ L e. Fil) -> (Fun (J fClusf L) <-> Fun {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))}))
11	10	3adant3 896	. . 3 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (Fun (J fClusf L) <-> Fun {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))}))
12	5, 11	mpbiri 211	. 2 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> Fun (J fClusf L))
13		simp3 878	. . . . 5 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> F:Y-->X)
14		eqid 1884	. . . . 5 |- ((fClus` J)` ((X FilMap L)` F)) = ((fClus` J)` ((X FilMap L)` F))
15	13, 14	jctir 317	. . . 4 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (F:Y-->X /\ ((fClus` J)` ((X FilMap L)` F)) = ((fClus` J)` ((X FilMap L)` F))))
16		fex 4595	. . . . . . . 8 |- ((F:Y-->X /\ Y e. _V) -> F e. _V)
17		uniexg 3795	. . . . . . . . 9 |- (L e. Fil -> U.L e. _V)
18	17, 7	syl5eqel 1975	. . . . . . . 8 |- (L e. Fil -> Y e. _V)
19	16, 18	sylan2 500	. . . . . . 7 |- ((F:Y-->X /\ L e. Fil) -> F e. _V)
20	19	ancoms 484	. . . . . 6 |- ((L e. Fil /\ F:Y-->X) -> F e. _V)
21	20	3adant1 894	. . . . 5 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> F e. _V)
22		fvex 4689	. . . . . 6 |- ((fClus` J)` ((X FilMap L)` F)) e. _V
23		feq1 4551	. . . . . . . 8 |- (f = F -> (f:Y-->X <-> F:Y-->X))
24		fveq2 4681	. . . . . . . . . 10 |- (f = F -> ((X FilMap L)` f) = ((X FilMap L)` F))
25	24	fveq2d 4685	. . . . . . . . 9 |- (f = F -> ((fClus` J)` ((X FilMap L)` f)) = ((fClus` J)` ((X FilMap L)` F)))
26	25	eqeq2d 1895	. . . . . . . 8 |- (f = F -> (s = ((fClus` J)` ((X FilMap L)` f)) <-> s = ((fClus` J)` ((X FilMap L)` F))))
27	23, 26	anbi12d 690	. . . . . . 7 |- (f = F -> ((f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f))) <-> (F:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` F)))))
28		eqeq1 1890	. . . . . . . 8 |- (s = ((fClus` J)` ((X FilMap L)` F)) -> (s = ((fClus` J)` ((X FilMap L)` F)) <-> ((fClus` J)` ((X FilMap L)` F)) = ((fClus` J)` ((X FilMap L)` F))))
29	28	anbi2d 678	. . . . . . 7 |- (s = ((fClus` J)` ((X FilMap L)` F)) -> ((F:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` F))) <-> (F:Y-->X /\ ((fClus` J)` ((X FilMap L)` F)) = ((fClus` J)` ((X FilMap L)` F)))))
30	27, 29	opelopabg 3567	. . . . . 6 |- ((F e. _V /\ ((fClus` J)` ((X FilMap L)` F)) e. _V) -> (<.F, ((fClus` J)` ((X FilMap L)` F))>. e. {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))} <-> (F:Y-->X /\ ((fClus` J)` ((X FilMap L)` F)) = ((fClus` J)` ((X FilMap L)` F)))))
31	22, 30	mpan2 760	. . . . 5 |- (F e. _V -> (<.F, ((fClus` J)` ((X FilMap L)` F))>. e. {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))} <-> (F:Y-->X /\ ((fClus` J)` ((X FilMap L)` F)) = ((fClus` J)` ((X FilMap L)` F)))))
32	21, 31	syl 12	. . . 4 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (<.F, ((fClus` J)` ((X FilMap L)` F))>. e. {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))} <-> (F:Y-->X /\ ((fClus` J)` ((X FilMap L)` F)) = ((fClus` J)` ((X FilMap L)` F)))))
33	15, 32	mpbird 213	. . 3 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> <.F, ((fClus` J)` ((X FilMap L)` F))>. e. {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))})
34	8	3adant3 896	. . 3 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (J fClusf L) = {<.f, s>. | (f:Y-->X /\ s = ((fClus` J)` ((X FilMap L)` f)))})
35	33, 34	eleqtrrd 1974	. 2 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> <.F, ((fClus` J)` ((X FilMap L)` F))>. e. (J fClusf L))
36	22	funopfv 4710	. 2 |- (Fun (J fClusf L) -> (<.F, ((fClus` J)` ((X FilMap L)` F))>. e. (J fClusf L) -> ((J fClusf L)` F) = ((fClus` J)` ((X FilMap L)` F))))
37	12, 35, 36	sylc 83	1 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> ((J fClusf L)` F) = ((fClus` J)` ((X FilMap L)` F)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  <.cop 3046  U.cuni 3177  {copab 3395  Fun wfun 3992  -->wf 3994  ` cfv 3998  (class class class)co 4884  Topctop 8857  Filcfil 10264   FilMap cfilmap 10304  fCluscfclus 15582   fClusf cfclusf 15583

This theorem is referenced by:  isfclusf 15625  flfssfcf 15629  uffcfflf 15630  fclsfcnp 15631

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

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