MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmeocnvb Structured version   Visualization version   GIF version

Theorem hmeocnvb 21798
Description: The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmeocnvb (Rel 𝐹 → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))

Proof of Theorem hmeocnvb
StepHypRef Expression
1 hmeocnv 21786 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾Homeo𝐽))
2 dfrel2 5724 . . . 4 (Rel 𝐹𝐹 = 𝐹)
3 eleq1 2838 . . . 4 (𝐹 = 𝐹 → (𝐹 ∈ (𝐾Homeo𝐽) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))
42, 3sylbi 207 . . 3 (Rel 𝐹 → (𝐹 ∈ (𝐾Homeo𝐽) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))
51, 4syl5ib 234 . 2 (Rel 𝐹 → (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾Homeo𝐽)))
6 hmeocnv 21786 . 2 (𝐹 ∈ (𝐾Homeo𝐽) → 𝐹 ∈ (𝐽Homeo𝐾))
75, 6impbid1 215 1 (Rel 𝐹 → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  ccnv 5248  Rel wrel 5254  (class class class)co 6793  Homeochmeo 21777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-top 20919  df-topon 20936  df-cn 21252  df-hmeo 21779
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator