hmeocnv - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > hmeocnv		Structured version Visualization version GIF version

Theorem hmeocnv 22913

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 22912	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
2		hmeocn 22911	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		eqid 2738	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2738	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 22397	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6		frel 6605	. . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Rel 𝐹)
7	2, 5, 6	3syl 18	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → Rel 𝐹)
8		dfrel2 6092	. . . 4 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
9	7, 8	sylib 217	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡^◡𝐹 = 𝐹)
10	9, 2	eqeltrd 2839	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡^◡𝐹 ∈ (𝐽 Cn 𝐾))
11		ishmeo 22910	. 2 ⊢ (^◡𝐹 ∈ (𝐾Homeo𝐽) ↔ (^◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ ^◡^◡𝐹 ∈ (𝐽 Cn 𝐾)))
12	1, 10, 11	sylanbrc 583	1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾Homeo𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∪ cuni 4839 ^◡ccnv 5588 Rel wrel 5594 ⟶wf 6429 (class class class)co 7275 Cn ccn 22375 Homeochmeo 22904

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-top 22043 df-topon 22060 df-cn 22378 df-hmeo 22906

This theorem is referenced by: hmeocnvb 22925 hmphsym 22933 xpstopnlem2 22962

W3C validator