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| Mirrors > Home > MPE Home > Th. List > idhmeo | Structured version Visualization version GIF version | ||
| Description: The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| idhmeo | ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽Homeo𝐽)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idcn 23266 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)) | |
| 2 | cnvresid 6644 | . . 3 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋) | |
| 3 | 2, 1 | eqeltrid 2844 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ◡( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)) | 
| 4 | ishmeo 23768 | . 2 ⊢ (( I ↾ 𝑋) ∈ (𝐽Homeo𝐽) ↔ (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽) ∧ ◡( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))) | |
| 5 | 1, 3, 4 | sylanbrc 583 | 1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽Homeo𝐽)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 I cid 5576 ◡ccnv 5683 ↾ cres 5686 ‘cfv 6560 (class class class)co 7432 TopOnctopon 22917 Cn ccn 23233 Homeochmeo 23762 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-top 22901 df-topon 22918 df-cn 23236 df-hmeo 23764 | 
| This theorem is referenced by: hmphref 23790 | 
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