Theorem flimfnei 10319

Description: The property of being a limit point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 9-Nov-2009.)

Hypotheses

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

Assertion

Ref	Expression
flimfnei	|- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (A e. ((J fLimf L)` F) <-> (A e. X /\ A.n e. ((nei` J)` {A})E.s e. L (F"s) C_ n)))

Distinct variable groups:   A,n   n,s,F   n,J   n,L,s   n,X,s   n,Y,s

Proof of Theorem flimfnei

Step	Hyp	Ref	Expression
1		flimfnei.1	. . . 4 |- X = U.J
2		flimfnei.2	. . . 4 |- Y = U.L
3	1, 2	sflimf 10318	. . 3 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> ((J fLimf L)` F) = ((fLim1` J)` ((X FilMap L)` F)))
4	3	eleq2d 1964	. 2 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (A e. ((J fLimf L)` F) <-> A e. ((fLim1` J)` ((X FilMap L)` F))))
5		simp1 876	. . 3 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> J e. Top)
6		uniexg 3795	. . . . . 6 |- (J e. Top -> U.J e. _V)
7	6, 1	syl5eqel 1975	. . . . 5 |- (J e. Top -> X e. _V)
8	7	3ad2ant1 897	. . . 4 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> X e. _V)
9		filfbas 10276	. . . . 5 |- (L e. Fil -> L e. fBas)
10	9	3ad2ant2 898	. . . 4 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> L e. fBas)
11		simp3 878	. . . 4 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> F:Y-->X)
12	2	fmf 10310	. . . 4 |- ((X e. _V /\ L e. fBas /\ F:Y-->X) -> ((X FilMap L)` F) e. Fil)
13	8, 10, 11, 12	syl111anc 1100	. . 3 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> ((X FilMap L)` F) e. Fil)
14	2	fmbas 10311	. . . . 5 |- ((X e. _V /\ L e. fBas /\ F:Y-->X) -> U.((X FilMap L)` F) = X)
15	8, 10, 11, 14	syl111anc 1100	. . . 4 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> U.((X FilMap L)` F) = X)
16	15	eqcomd 1889	. . 3 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> X = U.((X FilMap L)` F))
17		eqid 1884	. . . 4 |- U.((X FilMap L)` F) = U.((X FilMap L)` F)
18	1, 17	isfillim 10298	. . 3 |- ((J e. Top /\ ((X FilMap L)` F) e. Fil /\ X = U.((X FilMap L)` F)) -> (A e. ((fLim1` J)` ((X FilMap L)` F)) <-> (A e. X /\ ((nei` J)` {A}) C_ ((X FilMap L)` F))))
19	5, 13, 16, 18	syl111anc 1100	. 2 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (A e. ((fLim1` J)` ((X FilMap L)` F)) <-> (A e. X /\ ((nei` J)` {A}) C_ ((X FilMap L)` F))))
20	1	neii1 8997	. . . . . . 7 |- ((J e. Top /\ n e. ((nei` J)` {A})) -> n C_ X)
21	20	3ad2antl1 1038	. . . . . 6 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ n e. ((nei` J)` {A})) -> n C_ X)
22	2	elfilmap 10312	. . . . . . . . 9 |- ((X e. _V /\ L e. fBas /\ F:Y-->X) -> (n e. ((X FilMap L)` F) <-> (n C_ X /\ E.s e. L (F"s) C_ n)))
23		id 73	. . . . . . . . 9 |- (F:Y-->X -> F:Y-->X)
24	22, 7, 9, 23	syl3an 1139	. . . . . . . 8 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (n e. ((X FilMap L)` F) <-> (n C_ X /\ E.s e. L (F"s) C_ n)))
25	24	adantr 425	. . . . . . 7 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ n C_ X) -> (n e. ((X FilMap L)` F) <-> (n C_ X /\ E.s e. L (F"s) C_ n)))
26		ibar 705	. . . . . . . 8 |- (n C_ X -> (E.s e. L (F"s) C_ n <-> (n C_ X /\ E.s e. L (F"s) C_ n)))
27	26	adantl 424	. . . . . . 7 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ n C_ X) -> (E.s e. L (F"s) C_ n <-> (n C_ X /\ E.s e. L (F"s) C_ n)))
28	25, 27	bitr4d 590	. . . . . 6 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ n C_ X) -> (n e. ((X FilMap L)` F) <-> E.s e. L (F"s) C_ n))
29	21, 28	syldan 516	. . . . 5 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ n e. ((nei` J)` {A})) -> (n e. ((X FilMap L)` F) <-> E.s e. L (F"s) C_ n))
30	29	ralbidva 2119	. . . 4 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (A.n e. ((nei` J)` {A})n e. ((X FilMap L)` F) <-> A.n e. ((nei` J)` {A})E.s e. L (F"s) C_ n))
31		dfss3 2611	. . . 4 |- (((nei` J)` {A}) C_ ((X FilMap L)` F) <-> A.n e. ((nei` J)` {A})n e. ((X FilMap L)` F))
32	30, 31	syl5bb 591	. . 3 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (((nei` J)` {A}) C_ ((X FilMap L)` F) <-> A.n e. ((nei` J)` {A})E.s e. L (F"s) C_ n))
33	32	anbi2d 678	. 2 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> ((A e. X /\ ((nei` J)` {A}) C_ ((X FilMap L)` F)) <-> (A e. X /\ A.n e. ((nei` J)` {A})E.s e. L (F"s) C_ n)))
34	4, 19, 33	3bitrd 603	1 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (A e. ((J fLimf L)` F) <-> (A e. X /\ A.n e. ((nei` J)` {A})E.s e. L (F"s) C_ n)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  {csn 3044  U.cuni 3177  "cima 3989  -->wf 3994  ` cfv 3998  (class class class)co 4884  Topctop 8857  neicnei 8988  fBascfbas 10257  Filcfil 10264  fLim1cflim1 10294   FilMap cfilmap 10304   fLimf cflimf 10305

This theorem is referenced by:  flimfneii 10320

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-nel 2020  df-ral 2109  df-rex 2110  df-rab 2112  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-iun 3257  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-nei 8989  df-fbas 10259  df-fg 10260  df-fil 10265  df-flim1 10295  df-filmap 10306  df-flimf 10316

Copyright terms: Public domain