Theorem isflimf 10323

Description: The property of being a limit point of a function. (Contributed by Jeff Hankins, 8-Nov-2009.)

Hypotheses

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

Assertion

Ref	Expression
isflimf	|- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (A e. ((J fLimf L)` F) <-> (A e. X /\ A.o e. J (A e. o -> E.s e. L (F"s) C_ o))))

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

Proof of Theorem isflimf

Step	Hyp	Ref	Expression
1		isflimf.1	. . . 4 |- X = U.J
2		isflimf.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		simpl 346	. . . 4 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. ((fLim1` J)` ((X FilMap L)` F))) -> (J e. Top /\ L e. Fil /\ F:Y-->X))
6		eqid 1884	. . . . . 6 |- U.((X FilMap L)` F) = U.((X FilMap L)` F)
7	1, 6	flimelbas 10300	. . . . 5 |- (((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)
8		simp1 876	. . . . . 6 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> J e. Top)
9		uniexg 3795	. . . . . . . . 9 |- (J e. Top -> U.J e. _V)
10	9, 1	syl5eqel 1975	. . . . . . . 8 |- (J e. Top -> X e. _V)
11	10	3ad2ant1 897	. . . . . . 7 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> X e. _V)
12		filfbas 10276	. . . . . . . 8 |- (L e. Fil -> L e. fBas)
13	12	3ad2ant2 898	. . . . . . 7 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> L e. fBas)
14		simp3 878	. . . . . . 7 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> F:Y-->X)
15	2	fmf 10310	. . . . . . 7 |- ((X e. _V /\ L e. fBas /\ F:Y-->X) -> ((X FilMap L)` F) e. Fil)
16	11, 13, 14, 15	syl111anc 1100	. . . . . 6 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> ((X FilMap L)` F) e. Fil)
17	2	fmbas 10311	. . . . . . . 8 |- ((X e. _V /\ L e. fBas /\ F:Y-->X) -> U.((X FilMap L)` F) = X)
18	17	eqcomd 1889	. . . . . . 7 |- ((X e. _V /\ L e. fBas /\ F:Y-->X) -> X = U.((X FilMap L)` F))
19	11, 13, 14, 18	syl111anc 1100	. . . . . 6 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> X = U.((X FilMap L)` F))
20	8, 16, 19	3jca 1050	. . . . 5 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (J e. Top /\ ((X FilMap L)` F) e. Fil /\ X = U.((X FilMap L)` F)))
21	7, 20	sylan 497	. . . 4 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. ((fLim1` J)` ((X FilMap L)` F))) -> A e. X)
22	5, 21	jca 310	. . 3 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. ((fLim1` J)` ((X FilMap L)` F))) -> ((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X))
23		simpl 346	. . . 4 |- ((A e. X /\ A.o e. J (A e. o -> o e. ((X FilMap L)` F))) -> A e. X)
24	23	anim2i 362	. . 3 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ (A e. X /\ A.o e. J (A e. o -> o e. ((X FilMap L)` F)))) -> ((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X))
25	1, 6	flimopn 10321	. . . . 5 |- (((J e. Top /\ ((X FilMap L)` F) e. Fil /\ X = U.((X FilMap L)` F)) /\ A e. X) -> (A e. ((fLim1` J)` ((X FilMap L)` F)) <-> A.o e. J (A e. o -> o e. ((X FilMap L)` F))))
26	25, 20	sylan 497	. . . 4 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (A e. ((fLim1` J)` ((X FilMap L)` F)) <-> A.o e. J (A e. o -> o e. ((X FilMap L)` F))))
27		ibar 705	. . . . 5 |- (A e. X -> (A.o e. J (A e. o -> o e. ((X FilMap L)` F)) <-> (A e. X /\ A.o e. J (A e. o -> o e. ((X FilMap L)` F)))))
28	27	adantl 424	. . . 4 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (A.o e. J (A e. o -> o e. ((X FilMap L)` F)) <-> (A e. X /\ A.o e. J (A e. o -> o e. ((X FilMap L)` F)))))
29	26, 28	bitrd 587	. . 3 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (A e. ((fLim1` J)` ((X FilMap L)` F)) <-> (A e. X /\ A.o e. J (A e. o -> o e. ((X FilMap L)` F)))))
30	22, 24, 29	pm5.21nd 744	. 2 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (A e. ((fLim1` J)` ((X FilMap L)` F)) <-> (A e. X /\ A.o e. J (A e. o -> o e. ((X FilMap L)` F)))))
31	2	elfilmap 10312	. . . . . . . 8 |- ((X e. _V /\ L e. fBas /\ F:Y-->X) -> (o e. ((X FilMap L)` F) <-> (o C_ X /\ E.s e. L (F"s) C_ o)))
32	11, 13, 14, 31	syl111anc 1100	. . . . . . 7 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (o e. ((X FilMap L)` F) <-> (o C_ X /\ E.s e. L (F"s) C_ o)))
33	32	adantr 425	. . . . . 6 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ o e. J) -> (o e. ((X FilMap L)` F) <-> (o C_ X /\ E.s e. L (F"s) C_ o)))
34		elssuni 3206	. . . . . . . . 9 |- (o e. J -> o C_ U.J)
35	34, 1	syl6ssr 2664	. . . . . . . 8 |- (o e. J -> o C_ X)
36	35	biantrurd 796	. . . . . . 7 |- (o e. J -> (E.s e. L (F"s) C_ o <-> (o C_ X /\ E.s e. L (F"s) C_ o)))
37	36	adantl 424	. . . . . 6 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ o e. J) -> (E.s e. L (F"s) C_ o <-> (o C_ X /\ E.s e. L (F"s) C_ o)))
38	33, 37	bitr4d 590	. . . . 5 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ o e. J) -> (o e. ((X FilMap L)` F) <-> E.s e. L (F"s) C_ o))
39	38	imbi2d 674	. . . 4 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ o e. J) -> ((A e. o -> o e. ((X FilMap L)` F)) <-> (A e. o -> E.s e. L (F"s) C_ o)))
40	39	ralbidva 2119	. . 3 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (A.o e. J (A e. o -> o e. ((X FilMap L)` F)) <-> A.o e. J (A e. o -> E.s e. L (F"s) C_ o)))
41	40	anbi2d 678	. 2 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> ((A e. X /\ A.o e. J (A e. o -> o e. ((X FilMap L)` F))) <-> (A e. X /\ A.o e. J (A e. o -> E.s e. L (F"s) C_ o))))
42	4, 30, 41	3bitrd 603	1 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (A e. ((J fLimf L)` F) <-> (A e. X /\ A.o e. J (A e. o -> E.s e. L (F"s) C_ o))))

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  U.cuni 3177  "cima 3989  -->wf 3994  ` cfv 3998  (class class class)co 4884  Topctop 8857  fBascfbas 10257  Filcfil 10264  fLim1cflim1 10294   FilMap cfilmap 10304   fLimf cflimf 10305

This theorem is referenced by:  flimfelbas 10324  conttnf 14944  flimfbas 15601

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

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