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Theorem hmeocnvb 22383
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 22371 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾Homeo𝐽))
2 dfrel2 6017 . . . 4 (Rel 𝐹𝐹 = 𝐹)
3 eleq1 2880 . . . 4 (𝐹 = 𝐹 → (𝐹 ∈ (𝐾Homeo𝐽) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))
42, 3sylbi 220 . . 3 (Rel 𝐹 → (𝐹 ∈ (𝐾Homeo𝐽) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))
51, 4syl5ib 247 . 2 (Rel 𝐹 → (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾Homeo𝐽)))
6 hmeocnv 22371 . 2 (𝐹 ∈ (𝐾Homeo𝐽) → 𝐹 ∈ (𝐽Homeo𝐾))
75, 6impbid1 228 1 (Rel 𝐹 → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2112  ccnv 5522  Rel wrel 5528  (class class class)co 7139  Homeochmeo 22362
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-top 21503  df-topon 21520  df-cn 21836  df-hmeo 22364
This theorem is referenced by: (None)
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