Theorem hmeocld 12520

Description: Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Hypothesis

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

Assertion

Ref	Expression
hmeocld	|- ( ( 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		hmeocnvcn 12514	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
2	1	adantr 274	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> `' F e. ( K Cn J ) )
3		imacnvcnv 5011	. . . . 5 |- ( `' `' F " A ) = ( F " A )
4		cnclima 12431	. . . . 5 |- ( ( `' F e. ( K Cn J ) /\ A e. ( Clsd ` J ) ) -> ( `' `' F " A ) e. ( Clsd ` K ) )
5	3, 4	eqeltrrid 2228	. . . 4 |- ( ( `' F e. ( K Cn J ) /\ A e. ( Clsd ` J ) ) -> ( F " A ) e. ( Clsd ` K ) )
6	5	ex 114	. . 3 |- ( `' F e. ( K Cn J ) -> ( A e. ( Clsd ` J ) -> ( F " A ) e. ( Clsd ` K ) ) )
7	2, 6	syl 14	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. ( Clsd ` J ) -> ( F " A ) e. ( Clsd ` K ) ) )
8		hmeocn 12513	. . . . 5 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
9	8	adantr 274	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> F e. ( J Cn K ) )
10		cnclima 12431	. . . . 5 |- ( ( F e. ( J Cn K ) /\ ( F " A ) e. ( Clsd ` K ) ) -> ( `' F " ( F " A ) ) e. ( Clsd ` J ) )
11	10	ex 114	. . . 4 |- ( F e. ( J Cn K ) -> ( ( F " A ) e. ( Clsd ` K ) -> ( `' F " ( F " A ) ) e. ( Clsd ` J ) ) )
12	9, 11	syl 14	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( F " A ) e. ( Clsd ` K ) -> ( `' F " ( F " A ) ) e. ( Clsd ` J ) ) )
13		hmeoopn.1	. . . . . . 7 |- X = U. J
14		eqid 2140	. . . . . . 7 |- U. K = U. K
15	13, 14	hmeof1o 12517	. . . . . 6 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> U. K )
16		f1of1 5374	. . . . . 6 |- ( F : X -1-1-onto-> U. K -> F : X -1-1-> U. K )
17	15, 16	syl 14	. . . . 5 |- ( F e. ( J Homeo K ) -> F : X -1-1-> U. K )
18		f1imacnv 5392	. . . . 5 |- ( ( F : X -1-1-> U. K /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
19	17, 18	sylan 281	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
20	19	eleq1d 2209	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( `' F " ( F " A ) ) e. ( Clsd ` J ) <-> A e. ( Clsd ` J ) ) )
21	12, 20	sylibd 148	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( F " A ) e. ( Clsd ` K ) -> A e. ( Clsd ` J ) ) )
22	7, 21	impbid 128	1 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. ( Clsd ` J ) <-> ( F " A ) e. ( Clsd ` K ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   C_ wss 3076   U.cuni 3744   `'ccnv 4546   "cima 4550   -1-1->wf1 5128   -1-1-onto->wf1o 5130   ` cfv 5131  (class class class)co 5782   Clsdccld 12300   Cn ccn 12393   Homeochmeo 12508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-map 6552  df-top 12204  df-topon 12217  df-cld 12303  df-cn 12396  df-hmeo 12509

This theorem is referenced by: (None)