Theorem mapdiscn 14871

Description: Any mapping whose domain is associated to the discrete topology is continuous.

Hypothesis

Ref	Expression
mapdiscn2.2	|- B = U.J

Assertion

Ref	Expression
mapdiscn	|- ((J e. Top /\ F:A-->B /\ A e. D) -> F e. (~PA Cn J))

Proof of Theorem mapdiscn

Step	Hyp	Ref	Expression
1		feq2 4552	. . . . . 6 |- (a = A -> (F:a-->B <-> F:A-->B))
2		pweq 3036	. . . . . . . 8 |- (a = A -> ~Pa = ~PA)
3	2	opreq1d 4897	. . . . . . 7 |- (a = A -> (~Pa Cn J) = (~PA Cn J))
4	3	eleq2d 1964	. . . . . 6 |- (a = A -> (F e. (~Pa Cn J) <-> F e. (~PA Cn J)))
5	1, 4	imbi12d 688	. . . . 5 |- (a = A -> ((F:a-->B -> F e. (~Pa Cn J)) <-> (F:A-->B -> F e. (~PA Cn J))))
6	5	imbi2d 674	. . . 4 |- (a = A -> ((J e. Top -> (F:a-->B -> F e. (~Pa Cn J))) <-> (J e. Top -> (F:A-->B -> F e. (~PA Cn J)))))
7		ffn 4562	. . . . . 6 |- (F:a-->B -> F Fn a)
8		visset 2295	. . . . . . 7 |- a e. _V
9		fnex 4535	. . . . . . . 8 |- ((F Fn a /\ a e. _V) -> F e. _V)
10		feq1 4551	. . . . . . . . . 10 |- (f = F -> (f:a-->B <-> F:a-->B))
11		eleq1 1957	. . . . . . . . . . 11 |- (f = F -> (f e. (~Pa Cn J) <-> F e. (~Pa Cn J)))
12	11	imbi2d 674	. . . . . . . . . 10 |- (f = F -> ((J e. Top -> f e. (~Pa Cn J)) <-> (J e. Top -> F e. (~Pa Cn J))))
13	10, 12	imbi12d 688	. . . . . . . . 9 |- (f = F -> ((f:a-->B -> (J e. Top -> f e. (~Pa Cn J))) <-> (F:a-->B -> (J e. Top -> F e. (~Pa Cn J)))))
14		visset 2295	. . . . . . . . . . 11 |- f e. _V
15		mapdiscn2.2	. . . . . . . . . . 11 |- B = U.J
16	14, 15, 8	mapdiscnlem 14870	. . . . . . . . . 10 |- ((J e. Top /\ f:a-->B) -> f e. (~Pa Cn J))
17	16	expcom 403	. . . . . . . . 9 |- (f:a-->B -> (J e. Top -> f e. (~Pa Cn J)))
18	13, 17	vtoclg 2346	. . . . . . . 8 |- (F e. _V -> (F:a-->B -> (J e. Top -> F e. (~Pa Cn J))))
19	9, 18	syl 12	. . . . . . 7 |- ((F Fn a /\ a e. _V) -> (F:a-->B -> (J e. Top -> F e. (~Pa Cn J))))
20	8, 19	mpan2 760	. . . . . 6 |- (F Fn a -> (F:a-->B -> (J e. Top -> F e. (~Pa Cn J))))
21	7, 20	mpcom 60	. . . . 5 |- (F:a-->B -> (J e. Top -> F e. (~Pa Cn J)))
22	21	com12 14	. . . 4 |- (J e. Top -> (F:a-->B -> F e. (~Pa Cn J)))
23	6, 22	vtoclg 2346	. . 3 |- (A e. D -> (J e. Top -> (F:A-->B -> F e. (~PA Cn J))))
24	23	com3l 38	. 2 |- (J e. Top -> (F:A-->B -> (A e. D -> F e. (~PA Cn J))))
25	24	3imp 1061	1 |- ((J e. Top /\ F:A-->B /\ A e. D) -> F e. (~PA Cn J))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  ~Pcpw 3032  U.cuni 3177   Fn wfn 3993  -->wf 3994  (class class class)co 4884  Topctop 8857   Cn ccn 9028

This theorem is referenced by:  usptoplem 14917  istopx 14918  prtoptop 14919

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-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-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-cn 9030

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