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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 |
Ref | Expression |
---|---|
mptpreima |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmpo.1 | . . . . . 6 | |
2 | df-mpt 3991 | . . . . . 6 | |
3 | 1, 2 | eqtri 2160 | . . . . 5 |
4 | 3 | cnveqi 4714 | . . . 4 |
5 | cnvopab 4940 | . . . 4 | |
6 | 4, 5 | eqtri 2160 | . . 3 |
7 | 6 | imaeq1i 4878 | . 2 |
8 | df-ima 4552 | . . 3 | |
9 | resopab 4863 | . . . . 5 | |
10 | 9 | rneqi 4767 | . . . 4 |
11 | ancom 264 | . . . . . . . . 9 | |
12 | anass 398 | . . . . . . . . 9 | |
13 | 11, 12 | bitri 183 | . . . . . . . 8 |
14 | 13 | exbii 1584 | . . . . . . 7 |
15 | 19.42v 1878 | . . . . . . . 8 | |
16 | df-clel 2135 | . . . . . . . . . 10 | |
17 | 16 | bicomi 131 | . . . . . . . . 9 |
18 | 17 | anbi2i 452 | . . . . . . . 8 |
19 | 15, 18 | bitri 183 | . . . . . . 7 |
20 | 14, 19 | bitri 183 | . . . . . 6 |
21 | 20 | abbii 2255 | . . . . 5 |
22 | rnopab 4786 | . . . . 5 | |
23 | df-rab 2425 | . . . . 5 | |
24 | 21, 22, 23 | 3eqtr4i 2170 | . . . 4 |
25 | 10, 24 | eqtri 2160 | . . 3 |
26 | 8, 25 | eqtri 2160 | . 2 |
27 | 7, 26 | eqtri 2160 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wex 1468 wcel 1480 cab 2125 crab 2420 copab 3988 cmpt 3989 ccnv 4538 crn 4540 cres 4541 cima 4542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-mpt 3991 df-xp 4545 df-rel 4546 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 |
This theorem is referenced by: mptiniseg 5033 dmmpt 5034 fmpt 5570 f1oresrab 5585 suppssfv 5978 suppssov1 5979 infrenegsupex 9389 infxrnegsupex 11032 txcnmpt 12442 txdis1cn 12447 imasnopn 12468 |
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