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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 22908 | . 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 2110 ◡ccnv 5589 (class class class)co 7271 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-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: hmeocnv 22911 hmeof1o2 22912 hmeof1o 22913 hmeoopn 22915 hmeocld 22916 hmeocls 22917 hmeontr 22918 hmeoimaf1o 22919 hmeores 22920 hmeoco 22921 hmeoqtop 22924 hmphen 22934 haushmphlem 22936 cmphmph 22937 connhmph 22938 reghmph 22942 nrmhmph 22943 txhmeo 22952 xpstopnlem1 22958 tgpconncompeqg 23261 tgpconncomp 23262 qustgpopn 23269 mbfimaopnlem 24817 mndpluscn 31872 hmeocldb 34519 |
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