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Theorem hmeocnv 13892

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 13891	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
2		hmeocn 13890	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		eqid 2177	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2177	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 13789	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6		frel 5372	. . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Rel 𝐹)
7	2, 5, 6	3syl 17	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → Rel 𝐹)
8		dfrel2 5081	. . . 4 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
9	7, 8	sylib 122	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡^◡𝐹 = 𝐹)
10	9, 2	eqeltrd 2254	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡^◡𝐹 ∈ (𝐽 Cn 𝐾))
11		ishmeo 13889	. 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 1353 ∈ wcel 2148 ∪ cuni 3811 ^◡ccnv 4627 Rel wrel 4633 ⟶wf 5214 (class class class)co 5877 Cn ccn 13770 Homeochmeo 13885

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-map 6652 df-top 13583 df-topon 13596 df-cn 13773 df-hmeo 13886

This theorem is referenced by: hmeocnvb 13903

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