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 21470

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 21469	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
2		hmeocn 21468	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		eqid 2626	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2626	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 20955	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6		frel 6009	. . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Rel 𝐹)
7	2, 5, 6	3syl 18	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → Rel 𝐹)
8		dfrel2 5546	. . . 4 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
9	7, 8	sylib 208	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡^◡𝐹 = 𝐹)
10	9, 2	eqeltrd 2704	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡^◡𝐹 ∈ (𝐽 Cn 𝐾))
11		ishmeo 21467	. 2 ⊢ (^◡𝐹 ∈ (𝐾Homeo𝐽) ↔ (^◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ ^◡^◡𝐹 ∈ (𝐽 Cn 𝐾)))
12	1, 10, 11	sylanbrc 697	1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾Homeo𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1480 ∈ wcel 1992 ∪ cuni 4407 ^◡ccnv 5078 Rel wrel 5084 ⟶wf 5846 (class class class)co 6605 Cn ccn 20933 Homeochmeo 21461

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1841 ax-6 1890 ax-7 1937 ax-8 1994 ax-9 2001 ax-10 2021 ax-11 2036 ax-12 2049 ax-13 2250 ax-ext 2606 ax-sep 4746 ax-nul 4754 ax-pow 4808 ax-pr 4872 ax-un 6903

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1038 df-tru 1483 df-ex 1702 df-nf 1707 df-sb 1883 df-eu 2478 df-mo 2479 df-clab 2613 df-cleq 2619 df-clel 2622 df-nfc 2756 df-ne 2797 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3193 df-sbc 3423 df-dif 3563 df-un 3565 df-in 3567 df-ss 3574 df-nul 3897 df-if 4064 df-pw 4137 df-sn 4154 df-pr 4156 df-op 4160 df-uni 4408 df-br 4619 df-opab 4679 df-mpt 4680 df-id 4994 df-xp 5085 df-rel 5086 df-cnv 5087 df-co 5088 df-dm 5089 df-rn 5090 df-res 5091 df-ima 5092 df-iota 5813 df-fun 5852 df-fn 5853 df-f 5854 df-fv 5858 df-ov 6608 df-oprab 6609 df-mpt2 6610 df-map 7805 df-top 20616 df-topon 20618 df-cn 20936 df-hmeo 21463

This theorem is referenced by: hmeocnvb 21482 hmphsym 21490 xpstopnlem2 21519

W3C validator