Theorem hmeocld 15900

Description: Homeomorphisms preserve closedness.

Hypothesis

Ref	Expression
hmeocld.1	|- X = U.J

Assertion

Ref	Expression
hmeocld	|- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> (A e. (Clsd` J) <-> (F"A) e. (Clsd` K)))

Proof of Theorem hmeocld

Step	Hyp	Ref	Expression
1		difss 2735	. . . . 5 |- (X \ A) C_ X
2		hmeocld.1	. . . . . 6 |- X = U.J
3	2	hmeoopn 15899	. . . . 5 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ (X \ A) C_ X)) -> ((X \ A) e. J <-> (F"(X \ A)) e. K))
4	1, 3	mpanr2 776	. . . 4 |- (((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) -> ((X \ A) e. J <-> (F"(X \ A)) e. K))
5	4	adantrr 431	. . 3 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> ((X \ A) e. J <-> (F"(X \ A)) e. K))
6		eqid 1884	. . . . . . . . . 10 |- U.K = U.K
7	2, 6	hmeomap 10236	. . . . . . . . 9 |- ((J e. Top /\ K e. Top) -> (F e. (J Homeo K) -> F:X-1-1-onto->U.K))
8	7	imp 377	. . . . . . . 8 |- (((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) -> F:X-1-1-onto->U.K)
9		dff1o3 4641	. . . . . . . . 9 |- (F:X-1-1-onto->U.K <-> (F:X-onto->U.K /\ Fun `'F))
10	9	simprbi 353	. . . . . . . 8 |- (F:X-1-1-onto->U.K -> Fun `'F)
11	8, 10	syl 12	. . . . . . 7 |- (((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) -> Fun `'F)
12	11	adantrr 431	. . . . . 6 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> Fun `'F)
13		imadif 4493	. . . . . 6 |- (Fun `'F -> (F"(X \ A)) = ((F"X) \ (F"A)))
14	12, 13	syl 12	. . . . 5 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> (F"(X \ A)) = ((F"X) \ (F"A)))
15		f1ofo 4643	. . . . . . . 8 |- (F:X-1-1-onto->U.K -> F:X-onto->U.K)
16		foima 4622	. . . . . . . 8 |- (F:X-onto->U.K -> (F"X) = U.K)
17	8, 15, 16	3syl 24	. . . . . . 7 |- (((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) -> (F"X) = U.K)
18	17	adantrr 431	. . . . . 6 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> (F"X) = U.K)
19	18	difeq1d 2725	. . . . 5 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> ((F"X) \ (F"A)) = (U.K \ (F"A)))
20	14, 19	eqtrd 1925	. . . 4 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> (F"(X \ A)) = (U.K \ (F"A)))
21	20	eleq1d 1963	. . 3 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> ((F"(X \ A)) e. K <-> (U.K \ (F"A)) e. K))
22	5, 21	bitrd 587	. 2 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> ((X \ A) e. J <-> (U.K \ (F"A)) e. K))
23	2	iscld2 8946	. . 3 |- ((J e. Top /\ A C_ X) -> (A e. (Clsd` J) <-> (X \ A) e. J))
24	23	ad2ant2rl 447	. 2 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> (A e. (Clsd` J) <-> (X \ A) e. J))
25		simplr 449	. . 3 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> K e. Top)
26		imassrn 4278	. . . . . . . 8 |- (F"A) C_ ran F
27	26	a1i 8	. . . . . . 7 |- (F:X-1-1-onto->U.K -> (F"A) C_ ran F)
28		forn 4620	. . . . . . . 8 |- (F:X-onto->U.K -> ran F = U.K)
29	15, 28	syl 12	. . . . . . 7 |- (F:X-1-1-onto->U.K -> ran F = U.K)
30	27, 29	sseqtrd 2653	. . . . . 6 |- (F:X-1-1-onto->U.K -> (F"A) C_ U.K)
31	30	adantr 425	. . . . 5 |- ((F:X-1-1-onto->U.K /\ A C_ X) -> (F"A) C_ U.K)
32	31, 8	sylan 497	. . . 4 |- ((((J e. Top /\ K e. Top) /\ F e. (J Homeo K)) /\ A C_ X) -> (F"A) C_ U.K)
33	32	anasss 488	. . 3 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> (F"A) C_ U.K)
34	6	iscld2 8946	. . 3 |- ((K e. Top /\ (F"A) C_ U.K) -> ((F"A) e. (Clsd` K) <-> (U.K \ (F"A)) e. K))
35	25, 33, 34	syl11anc 524	. 2 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> ((F"A) e. (Clsd` K) <-> (U.K \ (F"A)) e. K))
36	22, 24, 35	3bitr4d 609	1 |- (((J e. Top /\ K e. Top) /\ (F e. (J Homeo K) /\ A C_ X)) -> (A e. (Clsd` J) <-> (F"A) e. (Clsd` K)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590   C_ wss 2593  U.cuni 3177  `'ccnv 3985  ran crn 3987  "cima 3989  Fun wfun 3992  -onto->wfo 3996  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  Topctop 8857  Clsdccld 8936   Homeo chomeosm 10230

This theorem is referenced by:  reheibor 16025

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-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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-cld 8939  df-homeo 10232

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