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Mirrors > Home > MPE Home > Th. List > mptpreima | Structured version Visualization version Unicode version |
Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
dmmpt.1 |
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Ref | Expression |
---|---|
mptpreima |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmpt.1 |
. . . . . 6
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2 | df-mpt 4456 |
. . . . . 6
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3 | 1, 2 | eqtri 2493 |
. . . . 5
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4 | 3 | cnveqi 5014 |
. . . 4
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5 | cnvopab 5243 |
. . . 4
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6 | 4, 5 | eqtri 2493 |
. . 3
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7 | 6 | imaeq1i 5171 |
. 2
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8 | df-ima 4852 |
. . 3
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9 | resopab 5157 |
. . . . 5
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10 | 9 | rneqi 5067 |
. . . 4
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11 | ancom 457 |
. . . . . . . . 9
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12 | anass 661 |
. . . . . . . . 9
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13 | 11, 12 | bitri 257 |
. . . . . . . 8
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14 | 13 | exbii 1726 |
. . . . . . 7
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15 | 19.42v 1842 |
. . . . . . . 8
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16 | df-clel 2467 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 16 | bicomi 207 |
. . . . . . . . 9
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18 | 17 | anbi2i 708 |
. . . . . . . 8
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19 | 15, 18 | bitri 257 |
. . . . . . 7
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20 | 14, 19 | bitri 257 |
. . . . . 6
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21 | 20 | abbii 2587 |
. . . . 5
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22 | rnopab 5085 |
. . . . 5
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23 | df-rab 2765 |
. . . . 5
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24 | 21, 22, 23 | 3eqtr4i 2503 |
. . . 4
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25 | 10, 24 | eqtri 2493 |
. . 3
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26 | 8, 25 | eqtri 2493 |
. 2
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27 | 7, 26 | eqtri 2493 |
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-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pr 4639 |
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-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-br 4396 df-opab 4455 df-mpt 4456 df-xp 4845 df-rel 4846 df-cnv 4847 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 |
This theorem is referenced by: mptiniseg 5336 dmmpt 5337 fmpt 6058 f1oresrab 6071 mptsuppdifd 6956 r0weon 8461 compss 8824 infrenegsup 10613 infmsupOLD 10614 eqglact 16946 odngen 17304 psrbagsn 18795 coe1mul2lem2 18938 pjdm 19347 xkoccn 20711 txcnmpt 20716 txdis1cn 20727 pthaus 20730 txkgen 20744 xkoco1cn 20749 xkoco2cn 20750 xkoinjcn 20779 txcon 20781 imasnopn 20782 imasncld 20783 imasncls 20784 ptcmplem1 21145 ptcmplem3 21147 ptcmplem4 21148 tmdgsum2 21189 symgtgp 21194 tgpconcompeqg 21204 ghmcnp 21207 tgpt0 21211 qustgpopn 21212 qustgphaus 21215 eltsms 21225 prdsxmslem2 21622 efopn 23682 atansopn 23937 xrlimcnp 23973 suppss2fOLD 28313 fpwrelmapffslem 28392 ptrest 32003 mbfposadd 32052 cnambfre 32053 itg2addnclem2 32058 iblabsnclem 32069 ftc1anclem1 32081 ftc1anclem6 32086 pwfi2f1o 36025 |
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