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Theorem hmeoima 19471
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 19467 . 2  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
2 imacnvcnv 5412 . . 3  |-  ( `' `' F " A )  =  ( F " A )
3 cnima 19002 . . 3  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  J )  ->  ( `' `' F " A )  e.  K
)
42, 3syl5eqelr 2547 . 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 1758   `'ccnv 4948   "cima 4952  (class class class)co 6201    Cn ccn 18961   Homeochmeo 19459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-map 7327  df-top 18636  df-topon 18639  df-cn 18964  df-hmeo 19461
This theorem is referenced by:  hmeoopn  19472  hmeoimaf1o  19476  hmeoqtop  19481  reghmph  19499  nrmhmph  19500  subgntr  19810  opnsubg  19811  tsmsxplem1  19860  tpr2rico  26488  cvmopnlem  27312
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