Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 22406 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)) | |
2 | cnvresid 6511 | . . 3 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋) | |
3 | 2, 1 | eqeltrid 2845 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ◡( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)) |
4 | ishmeo 22908 | . 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 2110 I cid 5489 ◡ccnv 5589 ↾ cres 5592 ‘cfv 6432 (class class class)co 7271 TopOnctopon 22057 Cn ccn 22373 Homeochmeo 22902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-map 8600 df-top 22041 df-topon 22058 df-cn 22376 df-hmeo 22904 |
This theorem is referenced by: hmphref 22930 |
Copyright terms: Public domain | W3C validator |