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Theorem hmeoima 20773
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 20769 . 2  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
2 imacnvcnv 5299 . . 3  |-  ( `' `' F " A )  =  ( F " A )
3 cnima 20274 . . 3  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  J )  ->  ( `' `' F " A )  e.  K
)
42, 3syl5eqelr 2533 . 2  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  J )  ->  ( F " A
)  e.  K )
51, 4sylan 474 1  |-  ( ( F  e.  ( J
Homeo K )  /\  A  e.  J )  ->  ( F " A )  e.  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    e. wcel 1886   `'ccnv 4832   "cima 4836  (class class class)co 6288    Cn ccn 20233   Homeochmeo 20761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-map 7471  df-top 19914  df-topon 19916  df-cn 20236  df-hmeo 20763
This theorem is referenced by:  hmeoopn  20774  hmeoimaf1o  20778  hmeoqtop  20783  reghmph  20801  nrmhmph  20802  subgntr  21114  opnsubg  21115  tsmsxplem1  21160  tpr2rico  28711  cvmopnlem  29994
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