Theorem flimff 10317

Description: Given a topology and a filtered set, return the convergence function on the functions from the filtered set to the base set of the topological space. (Contributed by Jeff Hankins, 14-Oct-2009.)

Hypotheses

Ref	Expression
flimff.1	|- X = U.J
flimff.2	|- Y = U.F

Assertion

Ref	Expression
flimff	|- ((J e. Top /\ F e. Fil) -> (J fLimf F) = {<.f, s>. | (f:Y-->X /\ s = ((fLim1` J)` ((X FilMap F)` f)))})

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

Proof of Theorem flimff

Step	Hyp	Ref	Expression
1		elmapg 5392	. . . . . 6 |- ((X e. _V /\ Y e. _V) -> (f e. (X ^m Y) <-> f:Y-->X))
2		uniexg 3795	. . . . . . 7 |- (J e. Top -> U.J e. _V)
3		flimff.1	. . . . . . 7 |- X = U.J
4	2, 3	syl5eqel 1975	. . . . . 6 |- (J e. Top -> X e. _V)
5		uniexg 3795	. . . . . . 7 |- (F e. Fil -> U.F e. _V)
6		flimff.2	. . . . . . 7 |- Y = U.F
7	5, 6	syl5eqel 1975	. . . . . 6 |- (F e. Fil -> Y e. _V)
8	1, 4, 7	syl2an 503	. . . . 5 |- ((J e. Top /\ F e. Fil) -> (f e. (X ^m Y) <-> f:Y-->X))
9	8	anbi1d 679	. . . 4 |- ((J e. Top /\ F e. Fil) -> ((f e. (X ^m Y) /\ s = ((fLim1` J)` ((X FilMap F)` f))) <-> (f:Y-->X /\ s = ((fLim1` J)` ((X FilMap F)` f)))))
10	9	opabbidv 3401	. . 3 |- ((J e. Top /\ F e. Fil) -> {<.f, s>. | (f e. (X ^m Y) /\ s = ((fLim1` J)` ((X FilMap F)` f)))} = {<.f, s>. | (f:Y-->X /\ s = ((fLim1` J)` ((X FilMap F)` f)))})
11		oprex 4907	. . . 4 |- (X ^m Y) e. _V
12	11	opabex2 4539	. . 3 |- {<.f, s>. | (f e. (X ^m Y) /\ s = ((fLim1` J)` ((X FilMap F)` f)))} e. _V
13	10, 12	syl6eqelr 1980	. 2 |- ((J e. Top /\ F e. Fil) -> {<.f, s>. | (f:Y-->X /\ s = ((fLim1` J)` ((X FilMap F)` f)))} e. _V)
14		unieq 3185	. . . . . . 7 |- (j = J -> U.j = U.J)
15	14, 3	syl6eqr 1946	. . . . . 6 |- (j = J -> U.j = X)
16		feq3 4553	. . . . . 6 |- (U.j = X -> (f:U.l-->U.j <-> f:U.l-->X))
17	15, 16	syl 12	. . . . 5 |- (j = J -> (f:U.l-->U.j <-> f:U.l-->X))
18		fveq2 4681	. . . . . . 7 |- (j = J -> (fLim1` j) = (fLim1` J))
19	15	opreq1d 4897	. . . . . . . 8 |- (j = J -> (U.j FilMap l) = (X FilMap l))
20	19	fveq1d 4683	. . . . . . 7 |- (j = J -> ((U.j FilMap l)` f) = ((X FilMap l)` f))
21	18, 20	fveq12d 10152	. . . . . 6 |- (j = J -> ((fLim1` j)` ((U.j FilMap l)` f)) = ((fLim1` J)` ((X FilMap l)` f)))
22	21	eqeq2d 1895	. . . . 5 |- (j = J -> (s = ((fLim1` j)` ((U.j FilMap l)` f)) <-> s = ((fLim1` J)` ((X FilMap l)` f))))
23	17, 22	anbi12d 690	. . . 4 |- (j = J -> ((f:U.l-->U.j /\ s = ((fLim1` j)` ((U.j FilMap l)` f))) <-> (f:U.l-->X /\ s = ((fLim1` J)` ((X FilMap l)` f)))))
24	23	opabbidv 3401	. . 3 |- (j = J -> {<.f, s>. | (f:U.l-->U.j /\ s = ((fLim1` j)` ((U.j FilMap l)` f)))} = {<.f, s>. | (f:U.l-->X /\ s = ((fLim1` J)` ((X FilMap l)` f)))})
25		unieq 3185	. . . . . . 7 |- (l = F -> U.l = U.F)
26	25, 6	syl6eqr 1946	. . . . . 6 |- (l = F -> U.l = Y)
27	26	feq2d 4557	. . . . 5 |- (l = F -> (f:U.l-->X <-> f:Y-->X))
28		opreq2 4890	. . . . . . . 8 |- (l = F -> (X FilMap l) = (X FilMap F))
29	28	fveq1d 4683	. . . . . . 7 |- (l = F -> ((X FilMap l)` f) = ((X FilMap F)` f))
30	29	fveq2d 4685	. . . . . 6 |- (l = F -> ((fLim1` J)` ((X FilMap l)` f)) = ((fLim1` J)` ((X FilMap F)` f)))
31	30	eqeq2d 1895	. . . . 5 |- (l = F -> (s = ((fLim1` J)` ((X FilMap l)` f)) <-> s = ((fLim1` J)` ((X FilMap F)` f))))
32	27, 31	anbi12d 690	. . . 4 |- (l = F -> ((f:U.l-->X /\ s = ((fLim1` J)` ((X FilMap l)` f))) <-> (f:Y-->X /\ s = ((fLim1` J)` ((X FilMap F)` f)))))
33	32	opabbidv 3401	. . 3 |- (l = F -> {<.f, s>. | (f:U.l-->X /\ s = ((fLim1` J)` ((X FilMap l)` f)))} = {<.f, s>. | (f:Y-->X /\ s = ((fLim1` J)` ((X FilMap F)` f)))})
34		df-flimf 10316	. . . 4 |- fLimf = {<.<.j, l>., z>. | (j e. Top /\ l e. Fil /\ z = {<.f, s>. | (f:U.l-->U.j /\ s = ((fLim1` j)` ((U.j FilMap l)` f)))})}
35		df-3an 860	. . . . 5 |- ((j e. Top /\ l e. Fil /\ z = {<.f, s>. | (f:U.l-->U.j /\ s = ((fLim1` j)` ((U.j FilMap l)` f)))}) <-> ((j e. Top /\ l e. Fil) /\ z = {<.f, s>. | (f:U.l-->U.j /\ s = ((fLim1` j)` ((U.j FilMap l)` f)))}))
36	35	oprabbii 4923	. . . 4 |- {<.<.j, l>., z>. | (j e. Top /\ l e. Fil /\ z = {<.f, s>. | (f:U.l-->U.j /\ s = ((fLim1` j)` ((U.j FilMap l)` f)))})} = {<.<.j, l>., z>. | ((j e. Top /\ l e. Fil) /\ z = {<.f, s>. | (f:U.l-->U.j /\ s = ((fLim1` j)` ((U.j FilMap l)` f)))})}
37	34, 36	eqtri 1908	. . 3 |- fLimf = {<.<.j, l>., z>. | ((j e. Top /\ l e. Fil) /\ z = {<.f, s>. | (f:U.l-->U.j /\ s = ((fLim1` j)` ((U.j FilMap l)` f)))})}
38	24, 33, 37	oprabval2g 4956	. 2 |- ((J e. Top /\ F e. Fil /\ {<.f, s>. | (f:Y-->X /\ s = ((fLim1` J)` ((X FilMap F)` f)))} e. _V) -> (J fLimf F) = {<.f, s>. | (f:Y-->X /\ s = ((fLim1` J)` ((X FilMap F)` f)))})
39	13, 38	mpd3an3 1192	1 |- ((J e. Top /\ F e. Fil) -> (J fLimf F) = {<.f, s>. | (f:Y-->X /\ s = ((fLim1` J)` ((X FilMap F)` f)))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  U.cuni 3177  {copab 3395  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885   ^m cmap 5381  Topctop 8857  Filcfil 10264  fLim1cflim1 10294   FilMap cfilmap 10304   fLimf cflimf 10305

This theorem is referenced by:  sflimf 10318

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-map 5383  df-flimf 10316

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