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Theorem hmeocnv 15298

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**
Step	Hyp	Ref	Expression
1		hmeocnvcn 15297	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
2		hmeocn 15296	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		eqid 2234	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2234	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 15195	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6		frel 5518	. . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Rel 𝐹)
7	2, 5, 6	3syl 17	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → Rel 𝐹)
8		dfrel2 5218	. . . 4 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
9	7, 8	sylib 122	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡^◡𝐹 = 𝐹)
10	9, 2	eqeltrd 2311	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡^◡𝐹 ∈ (𝐽 Cn 𝐾))
11		ishmeo 15295	. 2 ⊢ (^◡𝐹 ∈ (𝐾Homeo𝐽) ↔ (^◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ ^◡^◡𝐹 ∈ (𝐽 Cn 𝐾)))
12	1, 10, 11	sylanbrc 417	1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾Homeo𝐽))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1398 ∈ wcel 2205 ∪ cuni 3919 ^◡ccnv 4753 Rel wrel 4759 ⟶wf 5353 (class class class)co 6058 Cn ccn 15176 Homeochmeo 15291

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-fv 5365 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-map 6897 df-top 14989 df-topon 15002 df-cn 15179 df-hmeo 15292

This theorem is referenced by: hmeocnvb 15309

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