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Mirrors > Home > MPE Home > Th. List > hmeocn | Structured version Visualization version GIF version |
Description: A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
hmeocn | ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishmeo 22297 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ◡ccnv 5548 (class class class)co 7145 Cn ccn 21762 Homeochmeo 22291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8398 df-top 21432 df-topon 21449 df-cn 21765 df-hmeo 22293 |
This theorem is referenced by: hmeocnv 22300 hmeof1o2 22301 hmeof1o 22302 hmeoopn 22304 hmeocld 22305 hmeocls 22306 hmeontr 22307 hmeoimaf1o 22308 hmeores 22309 hmeoco 22310 hmeoqtop 22313 hmphen 22323 haushmphlem 22325 cmphmph 22326 connhmph 22327 reghmph 22331 nrmhmph 22332 txhmeo 22341 xpstopnlem1 22347 tgpconncompeqg 22649 tgpconncomp 22650 qustgpopn 22657 mbfimaopnlem 24185 mndpluscn 31069 hmeocldb 33580 |
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