Theorem dnsconst 7727

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 (_ (`'F"{P}) means "F is constant on A" (see funconstss 3793).

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 (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> F:X-->{P})

Proof of Theorem dnsconst

Step	Hyp	Ref	Expression
1		dnsconst.1	. . . . . . 7 |- X = U.J
2		dnsconst.2	. . . . . . 7 |- Y = U.K
3	1, 2	cnf 7701	. . . . . 6 |- ((J e. Top /\ K e. Top /\ F e. (J Cn K)) -> F:X-->Y)
4		haustop 7725	. . . . . 6 |- (K e. Haus -> K e. Top)
5	3, 4	syl3an2 858	. . . . 5 |- ((J e. Top /\ K e. Haus /\ F e. (J Cn K)) -> F:X-->Y)
6		ffn 3613	. . . . 5 |- (F:X-->Y -> F Fn X)
7	5, 6	syl 10	. . . 4 |- ((J e. Top /\ K e. Haus /\ F e. (J Cn K)) -> F Fn X)
8	7	adantr 389	. . 3 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> F Fn X)
9	2	sncld 7726	. . . . . . 7 |- ((K e. Haus /\ P e. Y) -> {P} e. (Clsd` K))
10	9	3ad2antl2 808	. . . . . 6 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ P e. Y) -> {P} e. (Clsd` K))
11		cnclima 7710	. . . . . . 7 |- (((J e. Top /\ K e. Top /\ F e. (J Cn K)) /\ {P} e. (Clsd` K)) -> (`'F"{P}) e. (Clsd` J))
12	11, 4	syl3anl2 871	. . . . . 6 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ {P} e. (Clsd` K)) -> (`'F"{P}) e. (Clsd` J))
13	10, 12	syldan 467	. . . . 5 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ P e. Y) -> (`'F"{P}) e. (Clsd` J))
14	13	3ad2antr1 810	. . . 4 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> (`'F"{P}) e. (Clsd` J))
15	1	clsss 7629	. . . . . . . . . . . 12 |- ((J e. Top /\ (`'F"{P}) (_ X /\ A (_ (`'F"{P})) -> ((cls` J)` A) (_ ((cls` J)` (`'F"{P})))
16		simpll 412	. . . . . . . . . . . 12 |- (((J e. Top /\ A (_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> J e. Top)
17	1	cldss 7613	. . . . . . . . . . . . 13 |- ((J e. Top /\ (`'F"{P}) e. (Clsd` J)) -> (`'F"{P}) (_ X)
18	17	adantlr 393	. . . . . . . . . . . 12 |- (((J e. Top /\ A (_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> (`'F"{P}) (_ X)
19		simplr 413	. . . . . . . . . . . 12 |- (((J e. Top /\ A (_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> A (_ (`'F"{P}))
20	15, 16, 18, 19	syl3anc 856	. . . . . . . . . . 11 |- (((J e. Top /\ A (_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> ((cls` J)` A) (_ ((cls` J)` (`'F"{P})))
21		cldcls 7624	. . . . . . . . . . . 12 |- ((J e. Top /\ (`'F"{P}) e. (Clsd` J)) -> ((cls` J)` (`'F"{P})) = (`'F"{P}))
22	21	adantlr 393	. . . . . . . . . . 11 |- (((J e. Top /\ A (_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> ((cls` J)` (`'F"{P})) = (`'F"{P}))
23	20, 22	sseqtrd 2087	. . . . . . . . . 10 |- (((J e. Top /\ A (_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> ((cls` J)` A) (_ (`'F"{P}))
24	23	ex 373	. . . . . . . . 9 |- ((J e. Top /\ A (_ (`'F"{P})) -> ((`'F"{P}) e. (Clsd` J) -> ((cls` J)` A) (_ (`'F"{P})))
25	24	adantrr 395	. . . . . . . 8 |- ((J e. Top /\ (A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> ((`'F"{P}) e. (Clsd` J) -> ((cls` J)` A) (_ (`'F"{P})))
26		sseq1 2072	. . . . . . . . 9 |- (((cls` J)` A) = X -> (((cls` J)` A) (_ (`'F"{P}) <-> X (_ (`'F"{P})))
27	26	ad2antll 407	. . . . . . . 8 |- ((J e. Top /\ (A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> (((cls` J)` A) (_ (`'F"{P}) <-> X (_ (`'F"{P})))
28	25, 27	sylibd 202	. . . . . . 7 |- ((J e. Top /\ (A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> ((`'F"{P}) e. (Clsd` J) -> X (_ (`'F"{P})))
29	28	adantlr 393	. . . . . 6 |- (((J e. Top /\ K e. Haus) /\ (A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> ((`'F"{P}) e. (Clsd` J) -> X (_ (`'F"{P})))
30	29	3adantr1 804	. . . . 5 |- (((J e. Top /\ K e. Haus) /\ (P e. Y /\ A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> ((`'F"{P}) e. (Clsd` J) -> X (_ (`'F"{P})))
31	30	3adantl3 803	. . . 4 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> ((`'F"{P}) e. (Clsd` J) -> X (_ (`'F"{P})))
32	14, 31	mpd 26	. . 3 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> X (_ (`'F"{P}))
33	8, 32	jca 288	. 2 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> (F Fn X /\ X (_ (`'F"{P})))
34		fconst3 3835	. 2 |- (F:X-->{P} <-> (F Fn X /\ X (_ (`'F"{P})))
35	33, 34	sylibr 200	1 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> F:X-->{P})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   (_ wss 2037  {csn 2399  U.cuni 2493  `'ccnv 3159  "cima 3163   Fn wfn 3167  -->wf 3168  ` cfv 3172  (class class class)co 3948  Topctop 7530  Clsdccld 7602  clsccl 7604   Cn ccn 7692  Hauscha 7720

This theorem is referenced by:  metdnsconst 7840

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-iin 2559  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fo 3186  df-fv 3188  df-opr 3950  df-oprab 3951  df-map 4308  df-top 7534  df-cld 7605  df-ntr 7606  df-cls 7607  df-cn 7694  df-haus 7721

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