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Mirrors > Home > ILE Home > Th. List > mptpreima | Unicode version |
Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
dmmpt2.1 |
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Ref | Expression |
---|---|
mptpreima |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmpt2.1 |
. . . . . 6
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2 | df-mpt 3811 |
. . . . . 6
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3 | 1, 2 | eqtri 2057 |
. . . . 5
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4 | 3 | cnveqi 4453 |
. . . 4
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5 | cnvopab 4669 |
. . . 4
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6 | 4, 5 | eqtri 2057 |
. . 3
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7 | 6 | imaeq1i 4608 |
. 2
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8 | df-ima 4301 |
. . 3
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9 | resopab 4595 |
. . . . 5
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10 | 9 | rneqi 4505 |
. . . 4
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11 | ancom 253 |
. . . . . . . . 9
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12 | anass 381 |
. . . . . . . . 9
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13 | 11, 12 | bitri 173 |
. . . . . . . 8
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14 | 13 | exbii 1493 |
. . . . . . 7
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15 | 19.42v 1783 |
. . . . . . . 8
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16 | df-clel 2033 |
. . . . . . . . . 10
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17 | 16 | bicomi 123 |
. . . . . . . . 9
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18 | 17 | anbi2i 430 |
. . . . . . . 8
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19 | 15, 18 | bitri 173 |
. . . . . . 7
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20 | 14, 19 | bitri 173 |
. . . . . 6
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21 | 20 | abbii 2150 |
. . . . 5
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22 | rnopab 4524 |
. . . . 5
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23 | df-rab 2309 |
. . . . 5
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24 | 21, 22, 23 | 3eqtr4i 2067 |
. . . 4
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25 | 10, 24 | eqtri 2057 |
. . 3
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26 | 8, 25 | eqtri 2057 |
. 2
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27 | 7, 26 | eqtri 2057 |
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-rab 2309 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-mpt 3811 df-xp 4294 df-rel 4295 df-cnv 4296 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 |
This theorem is referenced by: mptiniseg 4758 dmmpt 4759 fmpt 5262 f1oresrab 5272 suppssfv 5650 suppssov1 5651 |
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