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Theorem hmeocnv 12481

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 12480	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
2		hmeocn 12479	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		eqid 2139	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2139	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 12378	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6		frel 5277	. . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Rel 𝐹)
7	2, 5, 6	3syl 17	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → Rel 𝐹)
8		dfrel2 4989	. . . 4 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
9	7, 8	sylib 121	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡^◡𝐹 = 𝐹)
10	9, 2	eqeltrd 2216	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡^◡𝐹 ∈ (𝐽 Cn 𝐾))
11		ishmeo 12478	. 2 ⊢ (^◡𝐹 ∈ (𝐾Homeo𝐽) ↔ (^◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ ^◡^◡𝐹 ∈ (𝐽 Cn 𝐾)))
12	1, 10, 11	sylanbrc 413	1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾Homeo𝐽))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∪ cuni 3736 ^◡ccnv 4538 Rel wrel 4544 ⟶wf 5119 (class class class)co 5774 Cn ccn 12359 Homeochmeo 12474

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-top 12170 df-topon 12183 df-cn 12362 df-hmeo 12475

This theorem is referenced by: hmeocnvb 12492

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