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Mirrors > Home > MPE Home > Th. List > hmeof1o | Structured version Visualization version Unicode version |
Description: A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.) |
Ref | Expression |
---|---|
hmeof1o.1 |
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hmeof1o.2 |
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Ref | Expression |
---|---|
hmeof1o |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocn 20852 |
. . 3
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2 | cntop1 20333 |
. . . . 5
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3 | hmeof1o.1 |
. . . . . 6
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4 | 3 | toptopon 20025 |
. . . . 5
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5 | 2, 4 | sylib 201 |
. . . 4
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6 | cntop2 20334 |
. . . . 5
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7 | hmeof1o.2 |
. . . . . 6
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8 | 7 | toptopon 20025 |
. . . . 5
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9 | 6, 8 | sylib 201 |
. . . 4
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10 | 5, 9 | jca 541 |
. . 3
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11 | 1, 10 | syl 17 |
. 2
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12 | hmeof1o2 20855 |
. . 3
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13 | 12 | 3expia 1233 |
. 2
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14 | 11, 13 | mpcom 36 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-sbc 3256 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-map 7492 df-top 19998 df-topon 20000 df-cn 20320 df-hmeo 20847 |
This theorem is referenced by: hmeoopn 20858 hmeocld 20859 hmeontr 20861 hmeoimaf1o 20862 hmeoqtop 20867 haushmphlem 20879 cmphmph 20880 conhmph 20881 reghmph 20885 nrmhmph 20886 hmphdis 20888 hmphen2 20891 cmphaushmeo 20892 txhmeo 20895 tpr2rico 28792 mndpluscn 28806 |
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