Theorem flimfcnp 15602

Description: Continuity of a function at a point in terms of its limits.

Hypotheses

Ref	Expression
flimfcnp.1	|- X = U.J
flimfcnp.2	|- Y = U.K

Assertion

Ref	Expression
flimfcnp	|- (((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) -> (F e. ((J CnP K)` A) <-> A.f e. Fil ((X = U.f /\ A e. ((fLim1` J)` f)) -> (F` A) e. ((K fLimf f)` F))))

Distinct variable groups:   A,f   f,F   f,J   f,K   f,X   f,Y

Proof of Theorem flimfcnp

Step	Hyp	Ref	Expression
1		flimfcnp.1	. . 3 |- X = U.J
2		flimfcnp.2	. . 3 |- Y = U.K
3		eqid 1884	. . 3 |- (filGen` ({x | E.s e. f x = (F"s)} u. {Y})) = (filGen` ({x | E.s e. f x = (F"s)} u. {Y}))
4	1, 2, 3	cnpfillim 15589	. 2 |- (((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) -> (F e. ((J CnP K)` A) <-> A.f e. Fil ((X = U.f /\ A e. ((fLim1` J)` f)) -> (F` A) e. ((fLim1` K)` (filGen` ({x | E.s e. f x = (F"s)} u. {Y}))))))
5		simpl2 880	. . . . . . . 8 |- (((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) -> K e. Top)
6	5	ad2antrr 440	. . . . . . 7 |- (((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ f e. Fil) /\ (X = U.f /\ A e. ((fLim1` J)` f))) -> K e. Top)
7		simplr 449	. . . . . . 7 |- (((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ f e. Fil) /\ (X = U.f /\ A e. ((fLim1` J)` f))) -> f e. Fil)
8		feq2 4552	. . . . . . . . . . 11 |- (X = U.f -> (F:X-->Y <-> F:U.f-->Y))
9	8	biimpac 462	. . . . . . . . . 10 |- ((F:X-->Y /\ X = U.f) -> F:U.f-->Y)
10	9	3ad2antl3 1040	. . . . . . . . 9 |- (((J e. Top /\ K e. Top /\ F:X-->Y) /\ X = U.f) -> F:U.f-->Y)
11	10	adantlr 429	. . . . . . . 8 |- ((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ X = U.f) -> F:U.f-->Y)
12	11	ad2ant2r 445	. . . . . . 7 |- (((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ f e. Fil) /\ (X = U.f /\ A e. ((fLim1` J)` f))) -> F:U.f-->Y)
13		eqid 1884	. . . . . . . 8 |- U.f = U.f
14	2, 13	sflimf 10318	. . . . . . 7 |- ((K e. Top /\ f e. Fil /\ F:U.f-->Y) -> ((K fLimf f)` F) = ((fLim1` K)` ((Y FilMap f)` F)))
15	6, 7, 12, 14	syl111anc 1100	. . . . . 6 |- (((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ f e. Fil) /\ (X = U.f /\ A e. ((fLim1` J)` f))) -> ((K fLimf f)` F) = ((fLim1` K)` ((Y FilMap f)` F)))
16		uniexg 3795	. . . . . . . . . . . 12 |- (K e. Top -> U.K e. _V)
17	16, 2	syl5eqel 1975	. . . . . . . . . . 11 |- (K e. Top -> Y e. _V)
18	17	3ad2ant2 898	. . . . . . . . . 10 |- ((J e. Top /\ K e. Top /\ F:X-->Y) -> Y e. _V)
19	18	adantr 425	. . . . . . . . 9 |- (((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) -> Y e. _V)
20	19	ad2antrr 440	. . . . . . . 8 |- (((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ f e. Fil) /\ (X = U.f /\ A e. ((fLim1` J)` f))) -> Y e. _V)
21		filfbas 10276	. . . . . . . . 9 |- (f e. Fil -> f e. fBas)
22	21	ad2antlr 441	. . . . . . . 8 |- (((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ f e. Fil) /\ (X = U.f /\ A e. ((fLim1` J)` f))) -> f e. fBas)
23	13	isfilmap 10308	. . . . . . . 8 |- ((Y e. _V /\ f e. fBas /\ F:U.f-->Y) -> ((Y FilMap f)` F) = (filGen` ({x | E.s e. f x = (F"s)} u. {Y})))
24	20, 22, 12, 23	syl111anc 1100	. . . . . . 7 |- (((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ f e. Fil) /\ (X = U.f /\ A e. ((fLim1` J)` f))) -> ((Y FilMap f)` F) = (filGen` ({x | E.s e. f x = (F"s)} u. {Y})))
25	24	fveq2d 4685	. . . . . 6 |- (((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ f e. Fil) /\ (X = U.f /\ A e. ((fLim1` J)` f))) -> ((fLim1` K)` ((Y FilMap f)` F)) = ((fLim1` K)` (filGen` ({x | E.s e. f x = (F"s)} u. {Y}))))
26	15, 25	eqtr2d 1926	. . . . 5 |- (((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ f e. Fil) /\ (X = U.f /\ A e. ((fLim1` J)` f))) -> ((fLim1` K)` (filGen` ({x | E.s e. f x = (F"s)} u. {Y}))) = ((K fLimf f)` F))
27	26	eleq2d 1964	. . . 4 |- (((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ f e. Fil) /\ (X = U.f /\ A e. ((fLim1` J)` f))) -> ((F` A) e. ((fLim1` K)` (filGen` ({x | E.s e. f x = (F"s)} u. {Y}))) <-> (F` A) e. ((K fLimf f)` F)))
28	27	pm5.74da 646	. . 3 |- ((((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) /\ f e. Fil) -> (((X = U.f /\ A e. ((fLim1` J)` f)) -> (F` A) e. ((fLim1` K)` (filGen` ({x | E.s e. f x = (F"s)} u. {Y})))) <-> ((X = U.f /\ A e. ((fLim1` J)` f)) -> (F` A) e. ((K fLimf f)` F))))
29	28	ralbidva 2119	. 2 |- (((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) -> (A.f e. Fil ((X = U.f /\ A e. ((fLim1` J)` f)) -> (F` A) e. ((fLim1` K)` (filGen` ({x | E.s e. f x = (F"s)} u. {Y})))) <-> A.f e. Fil ((X = U.f /\ A e. ((fLim1` J)` f)) -> (F` A) e. ((K fLimf f)` F))))
30	4, 29	bitrd 587	1 |- (((J e. Top /\ K e. Top /\ F:X-->Y) /\ A e. X) -> (F e. ((J CnP K)` A) <-> A.f e. Fil ((X = U.f /\ A e. ((fLim1` J)` f)) -> (F` A) e. ((K fLimf f)` F))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591  {csn 3044  U.cuni 3177  "cima 3989  -->wf 3994  ` cfv 3998  (class class class)co 4884  Topctop 8857   CnP ccnp 9029  fBascfbas 10257  filGencfg 10258  Filcfil 10264  fLim1cflim1 10294   FilMap cfilmap 10304   fLimf cflimf 10305

This theorem is referenced by:  flimfcn 15603

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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