Theorem hmeocld 15123

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 15117	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
2	1	adantr 276	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> `' F e. ( K Cn J ) )
3		imacnvcnv 5208	. . . . 5 |- ( `' `' F " A ) = ( F " A )
4		cnclima 15034	. . . . 5 |- ( ( `' F e. ( K Cn J ) /\ A e. ( Clsd ` J ) ) -> ( `' `' F " A ) e. ( Clsd ` K ) )
5	3, 4	eqeltrrid 2319	. . . 4 |- ( ( `' F e. ( K Cn J ) /\ A e. ( Clsd ` J ) ) -> ( F " A ) e. ( Clsd ` K ) )
6	5	ex 115	. . 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 15116	. . . . 5 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
9	8	adantr 276	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> F e. ( J Cn K ) )
10		cnclima 15034	. . . . 5 |- ( ( F e. ( J Cn K ) /\ ( F " A ) e. ( Clsd ` K ) ) -> ( `' F " ( F " A ) ) e. ( Clsd ` J ) )
11	10	ex 115	. . . 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 2231	. . . . . . 7 |- U. K = U. K
15	13, 14	hmeof1o 15120	. . . . . 6 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> U. K )
16		f1of1 5591	. . . . . 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 5609	. . . . 5 |- ( ( F : X -1-1-> U. K /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
19	17, 18	sylan 283	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
20	19	eleq1d 2300	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( `' F " ( F " A ) ) e. ( Clsd ` J ) <-> A e. ( Clsd ` J ) ) )
21	12, 20	sylibd 149	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( F " A ) e. ( Clsd ` K ) -> A e. ( Clsd ` J ) ) )
22	7, 21	impbid 129	1 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. ( Clsd ` J ) <-> ( F " A ) e. ( Clsd ` K ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   C_ wss 3201   U.cuni 3898   `'ccnv 4730   "cima 4734   -1-1->wf1 5330   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   Clsdccld 14903   Cn ccn 14996   Homeochmeo 15111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-top 14809  df-topon 14822  df-cld 14906  df-cn 14999  df-hmeo 15112

This theorem is referenced by: (None)