Theorem hmeocld 12699

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 12693	. . . 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 5049	. . . . 5 |- ( `' `' F " A ) = ( F " A )
4		cnclima 12610	. . . . 5 |- ( ( `' F e. ( K Cn J ) /\ A e. ( Clsd ` J ) ) -> ( `' `' F " A ) e. ( Clsd ` K ) )
5	3, 4	eqeltrrid 2245	. . . 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 12692	. . . . 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 12610	. . . . 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 2157	. . . . . . 7 |- U. K = U. K
15	13, 14	hmeof1o 12696	. . . . . 6 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> U. K )
16		f1of1 5412	. . . . . 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 5430	. . . . 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 2226	. . 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 1335   e. wcel 2128   C_ wss 3102   U.cuni 3772   `'ccnv 4584   "cima 4588   -1-1->wf1 5166   -1-1-onto->wf1o 5168   ` cfv 5169  (class class class)co 5821   Clsdccld 12479   Cn ccn 12572   Homeochmeo 12687

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-map 6592  df-top 12383  df-topon 12396  df-cld 12482  df-cn 12575  df-hmeo 12688

This theorem is referenced by: (None)