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Mirrors > Home > MPE Home > Th. List > hmeocnvcn | Structured version Visualization version GIF version |
Description: The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
hmeocnvcn | ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishmeo 21933 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) | |
2 | 1 | simprbi 492 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 ◡ccnv 5341 (class class class)co 6905 Cn ccn 21399 Homeochmeo 21927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-map 8124 df-top 21069 df-topon 21086 df-cn 21402 df-hmeo 21929 |
This theorem is referenced by: hmeocnv 21936 hmeof1o2 21937 hmeoima 21939 hmeocld 21941 hmeocls 21942 hmeontr 21943 hmeores 21945 hmeoco 21946 txhmeo 21977 tgpconncompeqg 22285 mndpluscn 30517 cvmliftlem8 31820 hmeoclda 32866 |
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