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Theorem hmeocnv 22373
Description: The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmeocnv (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾Homeo𝐽))

Proof of Theorem hmeocnv
StepHypRef Expression
1 hmeocnvcn 22372 . 2 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
2 hmeocn 22371 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
3 eqid 2824 . . . . . 6 𝐽 = 𝐽
4 eqid 2824 . . . . . 6 𝐾 = 𝐾
53, 4cnf 21857 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
6 frel 6508 . . . . 5 (𝐹: 𝐽 𝐾 → Rel 𝐹)
72, 5, 63syl 18 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → Rel 𝐹)
8 dfrel2 6033 . . . 4 (Rel 𝐹𝐹 = 𝐹)
97, 8sylib 221 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 = 𝐹)
109, 2eqeltrd 2916 . 2 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
11 ishmeo 22370 . 2 (𝐹 ∈ (𝐾Homeo𝐽) ↔ (𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)))
121, 10, 11sylanbrc 586 1 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115   cuni 4824  ccnv 5541  Rel wrel 5547  wf 6339  (class class class)co 7149   Cn ccn 21835  Homeochmeo 22364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-top 21505  df-topon 21522  df-cn 21838  df-hmeo 22366
This theorem is referenced by:  hmeocnvb  22385  hmphsym  22393  xpstopnlem2  22422
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