Theorem sfvlim 10296

Description: Functions whose values are the limits of the filters. (Contributed by FL, 1-Sep-2008.)

Hypothesis

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

Assertion

Ref	Expression
sfvlim	|- (J e. Top -> (fLim1` J) = {<.a, b>. | (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})})

Distinct variable groups:   J,a,b,l   X,a,b,l

Proof of Theorem sfvlim

Step	Hyp	Ref	Expression
1		elin 2786	. . . . . . . . . . 11 |- (a e. (Fil i^i ~P~PX) <-> (a e. Fil /\ a e. ~P~PX))
2		simpll 448	. . . . . . . . . . 11 |- (((a e. Fil /\ U.a = X) /\ J e. Top) -> a e. Fil)
3		eqimss 2665	. . . . . . . . . . . . 13 |- (U.a = X -> U.a C_ X)
4		sspwuni 3333	. . . . . . . . . . . . . 14 |- (a C_ ~PX <-> U.a C_ X)
5		visset 2295	. . . . . . . . . . . . . . . 16 |- a e. _V
6	5	elpw 3037	. . . . . . . . . . . . . . 15 |- (a e. ~P~PX <-> a C_ ~PX)
7	6	biimpri 169	. . . . . . . . . . . . . 14 |- (a C_ ~PX -> a e. ~P~PX)
8	4, 7	sylbir 218	. . . . . . . . . . . . 13 |- (U.a C_ X -> a e. ~P~PX)
9	3, 8	syl 12	. . . . . . . . . . . 12 |- (U.a = X -> a e. ~P~PX)
10	9	ad2antlr 441	. . . . . . . . . . 11 |- (((a e. Fil /\ U.a = X) /\ J e. Top) -> a e. ~P~PX)
11	1, 2, 10	sylanbrc 527	. . . . . . . . . 10 |- (((a e. Fil /\ U.a = X) /\ J e. Top) -> a e. (Fil i^i ~P~PX))
12	11	ex 402	. . . . . . . . 9 |- ((a e. Fil /\ U.a = X) -> (J e. Top -> a e. (Fil i^i ~P~PX)))
13	12	3adant3 896	. . . . . . . 8 |- ((a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a}) -> (J e. Top -> a e. (Fil i^i ~P~PX)))
14	13	impcom 378	. . . . . . 7 |- ((J e. Top /\ (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})) -> a e. (Fil i^i ~P~PX))
15		simp3 878	. . . . . . . 8 |- ((a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a}) -> b = {l e. X | ((nei` J)` {l}) C_ a})
16	15	adantl 424	. . . . . . 7 |- ((J e. Top /\ (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})) -> b = {l e. X | ((nei` J)` {l}) C_ a})
17	14, 16	jca 310	. . . . . 6 |- ((J e. Top /\ (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})) -> (a e. (Fil i^i ~P~PX) /\ b = {l e. X | ((nei` J)` {l}) C_ a}))
18	17	ex 402	. . . . 5 |- (J e. Top -> ((a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a}) -> (a e. (Fil i^i ~P~PX) /\ b = {l e. X | ((nei` J)` {l}) C_ a})))
19	18	19.21aivv 1665	. . . 4 |- (J e. Top -> A.aA.b((a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a}) -> (a e. (Fil i^i ~P~PX) /\ b = {l e. X | ((nei` J)` {l}) C_ a})))
20		ssopab2 3573	. . . 4 |- ({<.a, b>. | (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})} C_ {<.a, b>. | (a e. (Fil i^i ~P~PX) /\ b = {l e. X | ((nei` J)` {l}) C_ a})} <-> A.aA.b((a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a}) -> (a e. (Fil i^i ~P~PX) /\ b = {l e. X | ((nei` J)` {l}) C_ a})))
21	19, 20	sylibr 217	. . 3 |- (J e. Top -> {<.a, b>. | (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})} C_ {<.a, b>. | (a e. (Fil i^i ~P~PX) /\ b = {l e. X | ((nei` J)` {l}) C_ a})})
22		uniexg 3795	. . . . . . 7 |- (J e. Top -> U.J e. _V)
23		sfvlim.1	. . . . . . 7 |- X = U.J
24	22, 23	syl5eqel 1975	. . . . . 6 |- (J e. Top -> X e. _V)
25		pwexg 3489	. . . . . 6 |- (X e. _V -> ~PX e. _V)
26		pwexg 3489	. . . . . 6 |- (~PX e. _V -> ~P~PX e. _V)
27	24, 25, 26	3syl 24	. . . . 5 |- (J e. Top -> ~P~PX e. _V)
28		inex1g 3454	. . . . 5 |- (~P~PX e. _V -> (~P~PX i^i Fil) e. _V)
29		incom 2787	. . . . . . 7 |- (~P~PX i^i Fil) = (Fil i^i ~P~PX)
30	29	eleq1i 1960	. . . . . 6 |- ((~P~PX i^i Fil) e. _V <-> (Fil i^i ~P~PX) e. _V)
31	30	biimpi 168	. . . . 5 |- ((~P~PX i^i Fil) e. _V -> (Fil i^i ~P~PX) e. _V)
32	27, 28, 31	3syl 24	. . . 4 |- (J e. Top -> (Fil i^i ~P~PX) e. _V)
33		opabex2g 4540	. . . 4 |- ((Fil i^i ~P~PX) e. _V -> {<.a, b>. | (a e. (Fil i^i ~P~PX) /\ b = {l e. X | ((nei` J)` {l}) C_ a})} e. _V)
34	32, 33	syl 12	. . 3 |- (J e. Top -> {<.a, b>. | (a e. (Fil i^i ~P~PX) /\ b = {l e. X | ((nei` J)` {l}) C_ a})} e. _V)
35		ssexg 3457	. . 3 |- (({<.a, b>. | (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})} C_ {<.a, b>. | (a e. (Fil i^i ~P~PX) /\ b = {l e. X | ((nei` J)` {l}) C_ a})} /\ {<.a, b>. | (a e. (Fil i^i ~P~PX) /\ b = {l e. X | ((nei` J)` {l}) C_ a})} e. _V) -> {<.a, b>. | (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})} e. _V)
36	21, 34, 35	syl11anc 524	. 2 |- (J e. Top -> {<.a, b>. | (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})} e. _V)
37		unieq 3185	. . . . . . 7 |- (x = J -> U.x = U.J)
38	37, 23	syl6eqr 1946	. . . . . 6 |- (x = J -> U.x = X)
39	38	eqeq2d 1895	. . . . 5 |- (x = J -> (U.a = U.x <-> U.a = X))
40		rabeq 2289	. . . . . . . 8 |- (U.x = X -> {l e. U.x | ((nei` x)` {l}) C_ a} = {l e. X | ((nei` x)` {l}) C_ a})
41	38, 40	syl 12	. . . . . . 7 |- (x = J -> {l e. U.x | ((nei` x)` {l}) C_ a} = {l e. X | ((nei` x)` {l}) C_ a})
42		fveq2 4681	. . . . . . . . . 10 |- (x = J -> (nei` x) = (nei` J))
43	42	fveq1d 4683	. . . . . . . . 9 |- (x = J -> ((nei` x)` {l}) = ((nei` J)` {l}))
44	43	sseq1d 2644	. . . . . . . 8 |- (x = J -> (((nei` x)` {l}) C_ a <-> ((nei` J)` {l}) C_ a))
45	44	rabbidv 2287	. . . . . . 7 |- (x = J -> {l e. X | ((nei` x)` {l}) C_ a} = {l e. X | ((nei` J)` {l}) C_ a})
46	41, 45	eqtrd 1925	. . . . . 6 |- (x = J -> {l e. U.x | ((nei` x)` {l}) C_ a} = {l e. X | ((nei` J)` {l}) C_ a})
47	46	eqeq2d 1895	. . . . 5 |- (x = J -> (b = {l e. U.x | ((nei` x)` {l}) C_ a} <-> b = {l e. X | ((nei` J)` {l}) C_ a}))
48	39, 47	3anbi23d 1171	. . . 4 |- (x = J -> ((a e. Fil /\ U.a = U.x /\ b = {l e. U.x | ((nei` x)` {l}) C_ a}) <-> (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})))
49	48	opabbidv 3401	. . 3 |- (x = J -> {<.a, b>. | (a e. Fil /\ U.a = U.x /\ b = {l e. U.x | ((nei` x)` {l}) C_ a})} = {<.a, b>. | (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})})
50		df-flim1 10295	. . 3 |- fLim1 = {<.x, y>. | (x e. Top /\ y = {<.a, b>. | (a e. Fil /\ U.a = U.x /\ b = {l e. U.x | ((nei` x)` {l}) C_ a})})}
51	49, 50	fvopab4g 4742	. 2 |- ((J e. Top /\ {<.a, b>. | (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})} e. _V) -> (fLim1` J) = {<.a, b>. | (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})})
52	36, 51	mpdan 768	1 |- (J e. Top -> (fLim1` J) = {<.a, b>. | (a e. Fil /\ U.a = X /\ b = {l e. X | ((nei` J)` {l}) C_ a})})

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  {crab 2108  _Vcvv 2292   i^i cin 2592   C_ wss 2593  ~Pcpw 3032  {csn 3044  U.cuni 3177  {copab 3395  ` cfv 3998  Topctop 8857  neicnei 8988  Filcfil 10264  fLim1cflim1 10294

This theorem is referenced by:  limfil 10297  limfillem1 14938

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-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  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-fv 4014  df-flim1 10295

Copyright terms: Public domain