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Mirrors > Home > MPE Home > Th. List > hmeoima | Structured version Visualization version Unicode version |
Description: The image of an open set by a homeomorphism is an open set. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
hmeoima |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocnvcn 20769 |
. 2
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2 | imacnvcnv 5299 |
. . 3
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3 | cnima 20274 |
. . 3
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4 | 2, 3 | syl5eqelr 2533 |
. 2
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5 | 1, 4 | sylan 474 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-rab 2745 df-v 3046 df-sbc 3267 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-fv 5589 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-map 7471 df-top 19914 df-topon 19916 df-cn 20236 df-hmeo 20763 |
This theorem is referenced by: hmeoopn 20774 hmeoimaf1o 20778 hmeoqtop 20783 reghmph 20801 nrmhmph 20802 subgntr 21114 opnsubg 21115 tsmsxplem1 21160 tpr2rico 28711 cvmopnlem 29994 |
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