Theorem dnsconst 7788

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 3808).

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 7762	. . . . . 6 |- ((J e. Top /\ K e. Top /\ F e. (J Cn K)) -> F:X-->Y)
4		haustop 7786	. . . . . 6 |- (K e. Haus -> K e. Top)
5	3, 4	syl3an2 860	. . . . 5 |- ((J e. Top /\ K e. Haus /\ F e. (J Cn K)) -> F:X-->Y)
6		ffn 3627	. . . . 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 7787	. . . . . . 7 |- ((K e. Haus /\ P e. Y) -> {P} e. (Clsd` K))
10	9	3ad2antl2 810	. . . . . 6 |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ P e. Y) -> {P} e. (Clsd` K))
11		cnclima 7771	. . . . . . 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 874	. . . . . 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 812	. . . 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 7687	. . . . . . . . . . . 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 7671	. . . . . . . . . . . . 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 858	. . . . . . . . . . 11 |- (((J e. Top /\ A (_ (`'F"{P})) /\ (`'F"{P}) e. (Clsd` J)) -> ((cls` J)` A) (_ ((cls` J)` (`'F"{P})))
21		cldcls 7682	. . . . . . . . . . . 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 2097	. . . . . . . . . 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 2082	. . . . . . . . 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 806	. . . . 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 805	. . . 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 3850	. 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 775   = wceq 956   e. wcel 958   (_ wss 2047  {csn 2409  U.cuni 2503  `'ccnv 3169  "cima 3173   Fn wfn 3177  -->wf 3178  ` cfv 3182  (class class class)co 3963  Topctop 7588  Clsdccld 7660  clsccl 7662   Cn ccn 7752  Hauscha 7781

This theorem is referenced by:  metdnsconst 7901

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-oprab 3966  df-map 4324  df-top 7592  df-cld 7663  df-ntr 7664  df-cls 7665  df-cn 7754  df-haus 7782

Copyright terms: Public domain