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Theorem hmeoima 20392
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 20388 . 2  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
2 imacnvcnv 5478 . . 3  |-  ( `' `' F " A )  =  ( F " A )
3 cnima 19893 . . 3  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  J )  ->  ( `' `' F " A )  e.  K
)
42, 3syl5eqelr 2550 . 2  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  J )  ->  ( F " A
)  e.  K )
51, 4sylan 471 1  |-  ( ( F  e.  ( J
Homeo K )  /\  A  e.  J )  ->  ( F " A )  e.  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819   `'ccnv 5007   "cima 5011  (class class class)co 6296    Cn ccn 19852   Homeochmeo 20380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-top 19526  df-topon 19529  df-cn 19855  df-hmeo 20382
This theorem is referenced by:  hmeoopn  20393  hmeoimaf1o  20397  hmeoqtop  20402  reghmph  20420  nrmhmph  20421  subgntr  20731  opnsubg  20732  tsmsxplem1  20781  tpr2rico  28055  cvmopnlem  28920
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