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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 22295 | . 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 5547 (class class class)co 7145 Cn ccn 21760 Homeochmeo 22289 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-top 21430 df-topon 21447 df-cn 21763 df-hmeo 22291 |
This theorem is referenced by: hmeocnv 22298 hmeof1o2 22299 hmeof1o 22300 hmeoopn 22302 hmeocld 22303 hmeocls 22304 hmeontr 22305 hmeoimaf1o 22306 hmeores 22307 hmeoco 22308 hmeoqtop 22311 hmphen 22321 haushmphlem 22323 cmphmph 22324 connhmph 22325 reghmph 22329 nrmhmph 22330 txhmeo 22339 xpstopnlem1 22345 tgpconncompeqg 22647 tgpconncomp 22648 qustgpopn 22655 mbfimaopnlem 24183 mndpluscn 31068 hmeocldb 33579 |
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