Theorem hmeoopn 14248

Description: Homeomorphisms preserve openness. (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
hmeoopn	|- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. J <-> ( F " A ) e. K ) )

Proof of Theorem hmeoopn

Step	Hyp	Ref	Expression
1		hmeoima 14247	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A e. J ) -> ( F " A ) e. K )
2	1	ex 115	. . 3 |- ( F e. ( J Homeo K ) -> ( A e. J -> ( F " A ) e. K ) )
3	2	adantr 276	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. J -> ( F " A ) e. K ) )
4		hmeocn 14242	. . . . 5 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
5		cnima 14157	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ ( F " A ) e. K ) -> ( `' F " ( F " A ) ) e. J )
6	5	ex 115	. . . . 5 |- ( F e. ( J Cn K ) -> ( ( F " A ) e. K -> ( `' F " ( F " A ) ) e. J ) )
7	4, 6	syl 14	. . . 4 |- ( F e. ( J Homeo K ) -> ( ( F " A ) e. K -> ( `' F " ( F " A ) ) e. J ) )
8	7	adantr 276	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( F " A ) e. K -> ( `' F " ( F " A ) ) e. J ) )
9		hmeoopn.1	. . . . . . 7 |- X = U. J
10		eqid 2189	. . . . . . 7 |- U. K = U. K
11	9, 10	hmeof1o 14246	. . . . . 6 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> U. K )
12		f1of1 5476	. . . . . 6 |- ( F : X -1-1-onto-> U. K -> F : X -1-1-> U. K )
13	11, 12	syl 14	. . . . 5 |- ( F e. ( J Homeo K ) -> F : X -1-1-> U. K )
14		f1imacnv 5494	. . . . 5 |- ( ( F : X -1-1-> U. K /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
15	13, 14	sylan 283	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
16	15	eleq1d 2258	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( `' F " ( F " A ) ) e. J <-> A e. J ) )
17	8, 16	sylibd 149	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( F " A ) e. K -> A e. J ) )
18	3, 17	impbid 129	1 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. J <-> ( F " A ) e. K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   C_ wss 3144   U.cuni 3824   `'ccnv 4640   "cima 4644   -1-1->wf1 5229   -1-1-onto->wf1o 5231  (class class class)co 5892   Cn ccn 14122   Homeochmeo 14237

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-map 6671  df-top 13935  df-topon 13948  df-cn 14125  df-hmeo 14238

This theorem is referenced by: (None)