Theorem hmeocnvb 21986

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

Step	Hyp	Ref	Expression
1		hmeocnv 21974	. . 3 ⊢ (◡𝐹 ∈ (𝐽Homeo𝐾) → ◡◡𝐹 ∈ (𝐾Homeo𝐽))
2		dfrel2 5837	. . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
3		eleq1 2846	. . . 4 ⊢ (◡◡𝐹 = 𝐹 → (◡◡𝐹 ∈ (𝐾Homeo𝐽) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))
4	2, 3	sylbi 209	. . 3 ⊢ (Rel 𝐹 → (◡◡𝐹 ∈ (𝐾Homeo𝐽) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))
5	1, 4	syl5ib 236	. 2 ⊢ (Rel 𝐹 → (◡𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾Homeo𝐽)))
6		hmeocnv 21974	. 2 ⊢ (𝐹 ∈ (𝐾Homeo𝐽) → ◡𝐹 ∈ (𝐽Homeo𝐾))
7	5, 6	impbid1 217	1 ⊢ (Rel 𝐹 → (◡𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1601   ∈ wcel 2106  ^◡ccnv 5354  Rel wrel 5360  (class class class)co 6922  Homeochmeo 21965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-top 21106  df-topon 21123  df-cn 21439  df-hmeo 21967

This theorem is referenced by: (None)