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Mirrors > Home > ILE Home > Th. List > dffo4 | Unicode version |
Description: Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.) |
Ref | Expression |
---|---|
dffo4 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffo2 5110 |
. . 3
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2 | simpl 102 |
. . . 4
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3 | vex 2560 |
. . . . . . . . . 10
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4 | 3 | elrn 4577 |
. . . . . . . . 9
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5 | eleq2 2101 |
. . . . . . . . 9
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6 | 4, 5 | syl5bbr 183 |
. . . . . . . 8
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7 | 6 | biimpar 281 |
. . . . . . 7
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8 | 7 | adantll 445 |
. . . . . 6
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9 | ffn 5046 |
. . . . . . . . . . 11
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10 | fnbr 5001 |
. . . . . . . . . . . 12
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11 | 10 | ex 108 |
. . . . . . . . . . 11
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12 | 9, 11 | syl 14 |
. . . . . . . . . 10
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13 | 12 | ancrd 309 |
. . . . . . . . 9
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14 | 13 | eximdv 1760 |
. . . . . . . 8
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15 | df-rex 2312 |
. . . . . . . 8
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16 | 14, 15 | syl6ibr 151 |
. . . . . . 7
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17 | 16 | ad2antrr 457 |
. . . . . 6
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18 | 8, 17 | mpd 13 |
. . . . 5
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19 | 18 | ralrimiva 2392 |
. . . 4
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20 | 2, 19 | jca 290 |
. . 3
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21 | 1, 20 | sylbi 114 |
. 2
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22 | fnbrfvb 5214 |
. . . . . . . . 9
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23 | 22 | biimprd 147 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | eqcom 2042 |
. . . . . . . 8
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25 | 23, 24 | syl6ib 150 |
. . . . . . 7
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26 | 9, 25 | sylan 267 |
. . . . . 6
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27 | 26 | reximdva 2421 |
. . . . 5
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28 | 27 | ralimdv 2388 |
. . . 4
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29 | 28 | imdistani 419 |
. . 3
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30 | dffo3 5314 |
. . 3
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31 | 29, 30 | sylibr 137 |
. 2
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32 | 21, 31 | impbii 117 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fo 4908 df-fv 4910 |
This theorem is referenced by: dffo5 5316 |
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