Theorem hmeoopn 15338

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 15337	. . . 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 15332	. . . . 5 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
5		cnima 15247	. . . . . 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 2238	. . . . . . 7 |- U. K = U. K
11	9, 10	hmeof1o 15336	. . . . . 6 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> U. K )
12		f1of1 5636	. . . . . 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 5654	. . . . 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 2307	. . 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 1402   e. wcel 2209   C_ wss 3220   U.cuni 3933   `'ccnv 4771   "cima 4775   -1-1->wf1 5372   -1-1-onto->wf1o 5374  (class class class)co 6078   Cn ccn 15212   Homeochmeo 15327

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-map 6917  df-top 15025  df-topon 15038  df-cn 15215  df-hmeo 15328

This theorem is referenced by: (None)