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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 |
---|---|
dmmpo.1 |
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Ref | Expression |
---|---|
mptpreima |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmpo.1 |
. . . . . 6
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2 | df-mpt 4063 |
. . . . . 6
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3 | 1, 2 | eqtri 2198 |
. . . . 5
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4 | 3 | cnveqi 4798 |
. . . 4
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5 | cnvopab 5026 |
. . . 4
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6 | 4, 5 | eqtri 2198 |
. . 3
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7 | 6 | imaeq1i 4963 |
. 2
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8 | df-ima 4636 |
. . 3
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9 | resopab 4947 |
. . . . 5
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10 | 9 | rneqi 4851 |
. . . 4
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11 | ancom 266 |
. . . . . . . . 9
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12 | anass 401 |
. . . . . . . . 9
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13 | 11, 12 | bitri 184 |
. . . . . . . 8
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14 | 13 | exbii 1605 |
. . . . . . 7
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15 | 19.42v 1906 |
. . . . . . . 8
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16 | df-clel 2173 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 16 | bicomi 132 |
. . . . . . . . 9
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18 | 17 | anbi2i 457 |
. . . . . . . 8
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19 | 15, 18 | bitri 184 |
. . . . . . 7
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20 | 14, 19 | bitri 184 |
. . . . . 6
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21 | 20 | abbii 2293 |
. . . . 5
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22 | rnopab 4870 |
. . . . 5
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23 | df-rab 2464 |
. . . . 5
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24 | 21, 22, 23 | 3eqtr4i 2208 |
. . . 4
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25 | 10, 24 | eqtri 2198 |
. . 3
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26 | 8, 25 | eqtri 2198 |
. 2
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27 | 7, 26 | eqtri 2198 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-mpt 4063 df-xp 4629 df-rel 4630 df-cnv 4631 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 |
This theorem is referenced by: mptiniseg 5119 dmmpt 5120 fmpt 5662 f1oresrab 5677 suppssfv 6073 suppssov1 6074 infrenegsupex 9583 infxrnegsupex 11255 txcnmpt 13440 txdis1cn 13445 imasnopn 13466 |
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