Theorem hmeocld 14548

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 14542	. . . 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 5134	. . . . 5 |- ( `' `' F " A ) = ( F " A )
4		cnclima 14459	. . . . 5 |- ( ( `' F e. ( K Cn J ) /\ A e. ( Clsd ` J ) ) -> ( `' `' F " A ) e. ( Clsd ` K ) )
5	3, 4	eqeltrrid 2284	. . . 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 14541	. . . . 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 14459	. . . . 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 2196	. . . . . . 7 |- U. K = U. K
15	13, 14	hmeof1o 14545	. . . . . 6 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> U. K )
16		f1of1 5503	. . . . . 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 5521	. . . . 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 2265	. . 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 1364   e. wcel 2167   C_ wss 3157   U.cuni 3839   `'ccnv 4662   "cima 4666   -1-1->wf1 5255   -1-1-onto->wf1o 5257   ` cfv 5258  (class class class)co 5922   Clsdccld 14328   Cn ccn 14421   Homeochmeo 14536

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-top 14234  df-topon 14247  df-cld 14331  df-cn 14424  df-hmeo 14537

This theorem is referenced by: (None)