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Mirrors > Home > ILE Home > Th. List > hmeocn | 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 13098 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) | |
2 | 1 | simplbi 272 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 ◡ccnv 4610 (class class class)co 5853 Cn ccn 12979 Homeochmeo 13094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-map 6628 df-top 12790 df-topon 12803 df-cn 12982 df-hmeo 13095 |
This theorem is referenced by: hmeocnv 13101 hmeof1o2 13102 hmeof1o 13103 hmeoopn 13105 hmeocld 13106 hmeontr 13107 hmeoimaf1o 13108 hmeores 13109 hmeoco 13110 txhmeo 13113 |
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