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Mirrors > Home > ILE Home > Th. List > f1ocnv | Unicode version |
Description: The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
f1ocnv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 4940 |
. . . . 5
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2 | dfrel2 4714 |
. . . . . 6
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3 | fneq1 4930 |
. . . . . . 7
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4 | 3 | biimprd 147 |
. . . . . 6
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5 | 2, 4 | sylbi 114 |
. . . . 5
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6 | 1, 5 | mpcom 32 |
. . . 4
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7 | 6 | anim2i 324 |
. . 3
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8 | 7 | ancoms 255 |
. 2
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9 | dff1o4 5077 |
. 2
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10 | dff1o4 5077 |
. 2
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11 | 8, 9, 10 | 3imtr4i 190 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 |
This theorem is referenced by: f1ocnvb 5083 f1orescnv 5085 f1imacnv 5086 f1cnv 5093 f1ococnv1 5098 f1oresrab 5272 f1ocnvfv2 5361 f1ocnvdm 5364 f1ocnvfvrneq 5365 fcof1o 5372 isocnv 5394 f1ofveu 5443 ener 6195 en0 6211 en1 6215 cnrecnv 9138 |
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