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Theorem hmeoima 21897
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 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝐽) → (𝐹𝐴) ∈ 𝐾)

Proof of Theorem hmeoima
StepHypRef Expression
1 hmeocnvcn 21893 . 2 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
2 imacnvcnv 5815 . . 3 (𝐹𝐴) = (𝐹𝐴)
3 cnima 21398 . . 3 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴𝐽) → (𝐹𝐴) ∈ 𝐾)
42, 3syl5eqelr 2883 . 2 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴𝐽) → (𝐹𝐴) ∈ 𝐾)
51, 4sylan 576 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝐽) → (𝐹𝐴) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  ccnv 5311  cima 5315  (class class class)co 6878   Cn ccn 21357  Homeochmeo 21885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-top 21027  df-topon 21044  df-cn 21360  df-hmeo 21887
This theorem is referenced by:  hmeoopn  21898  hmeoimaf1o  21902  hmeoqtop  21907  reghmph  21925  nrmhmph  21926  subgntr  22238  opnsubg  22239  tsmsxplem1  22284  tpr2rico  30474  cvmopnlem  31777
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