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Theorem hmeoima 17720
Description: The image of an open set by a homeomorphism is an open set. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmeoima  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  e.  J )  ->  ( F " A )  e.  K )

Proof of Theorem hmeoima
StepHypRef Expression
1 hmeocnvcn 17716 . 2  |-  ( F  e.  ( J  Homeo  K )  ->  `' F  e.  ( K  Cn  J
) )
2 imacnvcnv 5276 . . 3  |-  ( `' `' F " A )  =  ( F " A )
3 cnima 17253 . . 3  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  J )  ->  ( `' `' F " A )  e.  K
)
42, 3syl5eqelr 2474 . 2  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  J )  ->  ( F " A
)  e.  K )
51, 4sylan 458 1  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  e.  J )  ->  ( F " A )  e.  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   `'ccnv 4819   "cima 4823  (class class class)co 6022    Cn ccn 17212    Homeo chmeo 17708
This theorem is referenced by:  hmeoopn  17721  hmeoimaf1o  17725  hmeoqtop  17730  reghmph  17748  nrmhmph  17749  subgntr  18059  opnsubg  18060  tsmsxplem1  18105  tpr2rico  24116  cvmopnlem  24746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-map 6958  df-top 16888  df-topon 16891  df-cn 17215  df-hmeo 17710
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