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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 23767 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ◡ccnv 5684 (class class class)co 7431 Cn ccn 23232 Homeochmeo 23761 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-top 22900 df-topon 22917 df-cn 23235 df-hmeo 23763 |
| This theorem is referenced by: hmeocnv 23770 hmeof1o2 23771 hmeof1o 23772 hmeoopn 23774 hmeocld 23775 hmeocls 23776 hmeontr 23777 hmeoimaf1o 23778 hmeores 23779 hmeoco 23780 hmeoqtop 23783 hmphen 23793 haushmphlem 23795 cmphmph 23796 connhmph 23797 reghmph 23801 nrmhmph 23802 txhmeo 23811 xpstopnlem1 23817 tgpconncompeqg 24120 tgpconncomp 24121 qustgpopn 24128 mbfimaopnlem 25690 mndpluscn 33925 hmeocldb 36335 |
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