Theorem dnsconst 9065

Description: If a continuous mapping to a Hausdorff space is constant on a dense subset, it is constant on the entire space. Note that ((cls` J)` A) = X means "A is dense in X " and A C_ (`'F"{P}) means "F is constant on A " (see funconstss 4781).

Hypotheses

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

Assertion

Ref	Expression
dnsconst	|- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A C_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> F:X-->{P})

Proof of Theorem dnsconst

Step	Hyp	Ref	Expression
1		fconst3 4826	. 2 |- (F:X-->{P} <-> (F Fn X /\ X C_ (`'F"{P})))
2		dnsconst.1	. . . . . 6 |- X = U.J
3		dnsconst.2	. . . . . 6 |- Y = U.K
4	2, 3	cnf 9038	. . . . 5 |- ((J e. Top /\ K e. Top /\ F e. (J Cn K)) -> F:X-->Y)
5		haustop 9063	. . . . 5 |- (K e. Haus -> K e. Top)
6	4, 5	syl3an2 1131	. . . 4 |- ((J e. Top /\ K e. Haus /\ F e. (J Cn K)) -> F:X-->Y)
7		ffn 4562	. . . 4 |- (F:X-->Y -> F Fn X)
8	6, 7	syl 12	. . 3 |- ((J e. Top /\ K e. Haus /\ F e. (J Cn K)) -> F Fn X)
9	8	adantr 425	. 2 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A C_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> F Fn X)
10	3	sncld 9064	. . . . . 6 |- ((K e. Haus /\ P e. Y) -> {P} e. (Clsd` K))
11	10	3ad2antl2 1039	. . . . 5 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ P e. Y) -> {P} e. (Clsd` K))
12		cnclima 9048	. . . . . 6 |- (((J e. Top /\ K e. Top /\ F e. (J Cn K)) /\ {P} e. (Clsd` K)) -> (`'F"{P}) e. (Clsd` J))
13	12, 5	syl3anl2 1146	. . . . 5 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ {P} e. (Clsd` K)) -> (`'F"{P}) e. (Clsd` J))
14	11, 13	syldan 516	. . . 4 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ P e. Y) -> (`'F"{P}) e. (Clsd` J))
15	14	3ad2antr1 1041	. . 3 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A C_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> (`'F"{P}) e. (Clsd` J))
16		simpll 448	. . . . . . . . . . 11 |- (((J e. Top /\ A C_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> J e. Top)
17	2	cldss 8947	. . . . . . . . . . . 12 |- ((J e. Top /\ (`'F"{P}) e. (Clsd` J)) -> (`'F"{P}) C_ X)
18	17	adantlr 429	. . . . . . . . . . 11 |- (((J e. Top /\ A C_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> (`'F"{P}) C_ X)
19		simplr 449	. . . . . . . . . . 11 |- (((J e. Top /\ A C_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> A C_ (`'F"{P}))
20	2	clsss 8963	. . . . . . . . . . 11 |- ((J e. Top /\ (`'F"{P}) C_ X /\ A C_ (`'F"{P})) -> ((cls` J)` A) C_ ((cls` J)` (`'F"{P})))
21	16, 18, 19, 20	syl111anc 1100	. . . . . . . . . 10 |- (((J e. Top /\ A C_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> ((cls` J)` A) C_ ((cls` J)` (`'F"{P})))
22		cldcls 8958	. . . . . . . . . . 11 |- ((J e. Top /\ (`'F"{P}) e. (Clsd` J)) -> ((cls` J)` (`'F"{P})) = (`'F"{P}))
23	22	adantlr 429	. . . . . . . . . 10 |- (((J e. Top /\ A C_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> ((cls` J)` (`'F"{P})) = (`'F"{P}))
24	21, 23	sseqtrd 2653	. . . . . . . . 9 |- (((J e. Top /\ A C_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> ((cls` J)` A) C_ (`'F"{P}))
25	24	ex 402	. . . . . . . 8 |- ((J e. Top /\ A C_ (`'F"{P})) -> ((`'F"{P}) e. (Clsd` J) -> ((cls` J)` A) C_ (`'F"{P})))
26	25	adantrr 431	. . . . . . 7 |- ((J e. Top /\ (A C_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> ((`'F"{P}) e. (Clsd` J) -> ((cls` J)` A) C_ (`'F"{P})))
27		sseq1 2637	. . . . . . . 8 |- (((cls` J)` A) = X -> (((cls` J)` A) C_ (`'F"{P}) <-> X C_ (`'F"{P})))
28	27	ad2antll 443	. . . . . . 7 |- ((J e. Top /\ (A C_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> (((cls` J)` A) C_ (`'F"{P}) <-> X C_ (`'F"{P})))
29	26, 28	sylibd 219	. . . . . 6 |- ((J e. Top /\ (A C_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> ((`'F"{P}) e. (Clsd` J) -> X C_ (`'F"{P})))
30	29	adantlr 429	. . . . 5 |- (((J e. Top /\ K e. Haus) /\ (A C_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> ((`'F"{P}) e. (Clsd` J) -> X C_ (`'F"{P})))
31	30	3adantr1 1035	. . . 4 |- (((J e. Top /\ K e. Haus) /\ (P e. Y /\ A C_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> ((`'F"{P}) e. (Clsd` J) -> X C_ (`'F"{P})))
32	31	3adantl3 1034	. . 3 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A C_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> ((`'F"{P}) e. (Clsd` J) -> X C_ (`'F"{P})))
33	15, 32	mpd 29	. 2 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A C_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> X C_ (`'F"{P}))
34	1, 9, 33	sylanbrc 527	1 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A C_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> F:X-->{P})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   C_ wss 2593  {csn 3044  U.cuni 3177  `'ccnv 3985  "cima 3989   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  Topctop 8857  Clsdccld 8936  clsccl 8938   Cn ccn 9028  Hauscha 9058

This theorem is referenced by:  metdnsconst 9179

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-int 3215  df-iun 3257  df-iin 3258  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-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941  df-cn 9030  df-haus 9059

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