Theorem conttnf2 14945

Description: F is continous at point A iff (F` A) is a limit of the image filter of the neighborhoods of A.

Hypotheses

Ref	Expression
conttnf2.1	|- L = ((nei` J)` {A})
conttnf2.2	|- X = U.K
conttnf2.3	|- Y = U.J

Assertion

Ref	Expression
conttnf2	|- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> (F e. ((J CnP K)` A) <-> (F` A) e. ((fLim1` K)` ((X FilMap L)` F))))

Proof of Theorem conttnf2

Step	Hyp	Ref	Expression
1		conttnf2.1	. . 3 |- L = ((nei` J)` {A})
2		conttnf2.2	. . 3 |- X = U.K
3		conttnf2.3	. . 3 |- Y = U.J
4	1, 2, 3	conttnf 14944	. 2 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> (F e. ((J CnP K)` A) <-> (F` A) e. ((K fLimf L)` F)))
5		simp2 877	. . . 4 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> K e. Top)
6		simp1 876	. . . . . 6 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> J e. Top)
7		snssi 3129	. . . . . . . 8 |- (A e. Y -> {A} C_ Y)
8	7	adantr 425	. . . . . . 7 |- ((A e. Y /\ F:Y-->X) -> {A} C_ Y)
9	8	3ad2ant3 899	. . . . . 6 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> {A} C_ Y)
10		snnzg 3118	. . . . . . . 8 |- (A e. Y -> {A} =/= (/))
11	10	adantr 425	. . . . . . 7 |- ((A e. Y /\ F:Y-->X) -> {A} =/= (/))
12	11	3ad2ant3 899	. . . . . 6 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> {A} =/= (/))
13	3	neifil 10302	. . . . . 6 |- ((J e. Top /\ {A} C_ Y /\ {A} =/= (/)) -> ((nei` J)` {A}) e. Fil)
14	6, 9, 12, 13	syl111anc 1100	. . . . 5 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> ((nei` J)` {A}) e. Fil)
15	14, 1	syl5eqel 1975	. . . 4 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> L e. Fil)
16		simp3r 905	. . . . 5 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> F:Y-->X)
17	3	unnei 9011	. . . . . . . 8 |- ((J e. Top /\ {A} C_ Y) -> U.((nei` J)` {A}) = Y)
18	6, 9, 17	syl11anc 524	. . . . . . 7 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> U.((nei` J)` {A}) = Y)
19	1	unieqi 3187	. . . . . . 7 |- U.L = U.((nei` J)` {A})
20	18, 19	syl5eq 1940	. . . . . 6 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> U.L = Y)
21	20	feq2d 4557	. . . . 5 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> (F:U.L-->X <-> F:Y-->X))
22	16, 21	mpbird 213	. . . 4 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> F:U.L-->X)
23		eqid 1884	. . . . 5 |- U.L = U.L
24	2, 23	sflimf 10318	. . . 4 |- ((K e. Top /\ L e. Fil /\ F:U.L-->X) -> ((K fLimf L)` F) = ((fLim1` K)` ((X FilMap L)` F)))
25	5, 15, 22, 24	syl111anc 1100	. . 3 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> ((K fLimf L)` F) = ((fLim1` K)` ((X FilMap L)` F)))
26	25	eleq2d 1964	. 2 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> ((F` A) e. ((K fLimf L)` F) <-> (F` A) e. ((fLim1` K)` ((X FilMap L)` F))))
27	4, 26	bitrd 587	1 |- ((J e. Top /\ K e. Top /\ (A e. Y /\ F:Y-->X)) -> (F e. ((J CnP K)` A) <-> (F` A) e. ((fLim1` K)` ((X FilMap L)` F))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  -->wf 3994  ` cfv 3998  (class class class)co 4884  Topctop 8857  neicnei 8988   CnP ccnp 9029  Filcfil 10264  fLim1cflim1 10294   FilMap cfilmap 10304   fLimf cflimf 10305

This theorem is referenced by:  cnpfillim4 14947

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-top 8861  df-nei 8989  df-cnp 9031  df-fbas 10259  df-fg 10260  df-fil 10265  df-flim1 10295  df-filmap 10306  df-flimf 10316

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