Theorem mapudiscn 14872

Description: Any mapping whose range is associated to the undiscrete topology is continuous.

Hypotheses

Ref	Expression
mapudiscn.2	|- A = U.J
mapudiscn.3	|- B e. _V

Assertion

Ref	Expression
mapudiscn	|- ((J e. Top /\ F:A-->B) -> F e. (J Cn {(/), B}))

Proof of Theorem mapudiscn

Step	Hyp	Ref	Expression
1		indistop 8918	. 2 |- {(/), B} e. Top
2		simpr 350	. . . 4 |- (((J e. Top /\ {(/), B} e. Top) /\ F:A-->B) -> F:A-->B)
3		imaeq2 4260	. . . . . . . . . . . 12 |- (x = (/) -> (`'F"x) = (`'F"(/)))
4	3	eleq1d 1963	. . . . . . . . . . 11 |- (x = (/) -> ((`'F"x) e. J <-> (`'F"(/)) e. J))
5		0opn 8870	. . . . . . . . . . . 12 |- (J e. Top -> (/) e. J)
6		ima0 4283	. . . . . . . . . . . 12 |- (`'F"(/)) = (/)
7	5, 6	syl5eqel 1975	. . . . . . . . . . 11 |- (J e. Top -> (`'F"(/)) e. J)
8	4, 7	syl5bir 227	. . . . . . . . . 10 |- (x = (/) -> (J e. Top -> (`'F"x) e. J))
9	8	adantrd 427	. . . . . . . . 9 |- (x = (/) -> ((J e. Top /\ F:A-->B) -> (`'F"x) e. J))
10		imaeq2 4260	. . . . . . . . . . 11 |- (x = B -> (`'F"x) = (`'F"B))
11	10	eleq1d 1963	. . . . . . . . . 10 |- (x = B -> ((`'F"x) e. J <-> (`'F"B) e. J))
12		fimacnv 4783	. . . . . . . . . . . 12 |- (F:A-->B -> (`'F"B) = A)
13		eleq1 1957	. . . . . . . . . . . . 13 |- ((`'F"B) = A -> ((`'F"B) e. J <-> A e. J))
14		mapudiscn.2	. . . . . . . . . . . . . 14 |- A = U.J
15	14	topopn 8871	. . . . . . . . . . . . 13 |- (J e. Top -> A e. J)
16	13, 15	syl5bir 227	. . . . . . . . . . . 12 |- ((`'F"B) = A -> (J e. Top -> (`'F"B) e. J))
17	12, 16	syl 12	. . . . . . . . . . 11 |- (F:A-->B -> (J e. Top -> (`'F"B) e. J))
18	17	impcom 378	. . . . . . . . . 10 |- ((J e. Top /\ F:A-->B) -> (`'F"B) e. J)
19	11, 18	syl5bir 227	. . . . . . . . 9 |- (x = B -> ((J e. Top /\ F:A-->B) -> (`'F"x) e. J))
20	9, 19	jaoi 368	. . . . . . . 8 |- ((x = (/) \/ x = B) -> ((J e. Top /\ F:A-->B) -> (`'F"x) e. J))
21	20	com12 14	. . . . . . 7 |- ((J e. Top /\ F:A-->B) -> ((x = (/) \/ x = B) -> (`'F"x) e. J))
22	21	adantlr 429	. . . . . 6 |- (((J e. Top /\ {(/), B} e. Top) /\ F:A-->B) -> ((x = (/) \/ x = B) -> (`'F"x) e. J))
23		visset 2295	. . . . . . 7 |- x e. _V
24	23	elpr 3061	. . . . . 6 |- (x e. {(/), B} <-> (x = (/) \/ x = B))
25	22, 24	syl5ib 223	. . . . 5 |- (((J e. Top /\ {(/), B} e. Top) /\ F:A-->B) -> (x e. {(/), B} -> (`'F"x) e. J))
26	25	r19.21aiv 2175	. . . 4 |- (((J e. Top /\ {(/), B} e. Top) /\ F:A-->B) -> A.x e. {(/), B} (`'F"x) e. J)
27	2, 26	jca 310	. . 3 |- (((J e. Top /\ {(/), B} e. Top) /\ F:A-->B) -> (F:A-->B /\ A.x e. {(/), B} (`'F"x) e. J))
28		0ex 3446	. . . . . . 7 |- (/) e. _V
29		mapudiscn.3	. . . . . . 7 |- B e. _V
30	28, 29	unipr 3191	. . . . . 6 |- U.{(/), B} = ((/) u. B)
31		uncom 2744	. . . . . 6 |- ((/) u. B) = (B u. (/))
32		un0 2896	. . . . . 6 |- (B u. (/)) = B
33	30, 31, 32	3eqtrri 1913	. . . . 5 |- B = U.{(/), B}
34	14, 33	iscn 9034	. . . 4 |- ((J e. Top /\ {(/), B} e. Top) -> (F e. (J Cn {(/), B}) <-> (F:A-->B /\ A.x e. {(/), B} (`'F"x) e. J)))
35	34	adantr 425	. . 3 |- (((J e. Top /\ {(/), B} e. Top) /\ F:A-->B) -> (F e. (J Cn {(/), B}) <-> (F:A-->B /\ A.x e. {(/), B} (`'F"x) e. J)))
36	27, 35	mpbird 213	. 2 |- (((J e. Top /\ {(/), B} e. Top) /\ F:A-->B) -> F e. (J Cn {(/), B}))
37	1, 36	mpanl2 771	1 |- ((J e. Top /\ F:A-->B) -> F e. (J Cn {(/), B}))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   u. cun 2591  (/)c0 2875  {cpr 3045  U.cuni 3177  `'ccnv 3985  "cima 3989  -->wf 3994  (class class class)co 4884  Topctop 8857   Cn ccn 9028

This theorem is referenced by:  extopgrp 14980

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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