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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 4039 | . . . . . 6 | |
3 | 1, 2 | eqtri 2185 | . . . . 5 |
4 | 3 | cnveqi 4773 | . . . 4 |
5 | cnvopab 4999 | . . . 4 | |
6 | 4, 5 | eqtri 2185 | . . 3 |
7 | 6 | imaeq1i 4937 | . 2 |
8 | df-ima 4611 | . . 3 | |
9 | resopab 4922 | . . . . 5 | |
10 | 9 | rneqi 4826 | . . . 4 |
11 | ancom 264 | . . . . . . . . 9 | |
12 | anass 399 | . . . . . . . . 9 | |
13 | 11, 12 | bitri 183 | . . . . . . . 8 |
14 | 13 | exbii 1592 | . . . . . . 7 |
15 | 19.42v 1893 | . . . . . . . 8 | |
16 | df-clel 2160 | . . . . . . . . . 10 | |
17 | 16 | bicomi 131 | . . . . . . . . 9 |
18 | 17 | anbi2i 453 | . . . . . . . 8 |
19 | 15, 18 | bitri 183 | . . . . . . 7 |
20 | 14, 19 | bitri 183 | . . . . . 6 |
21 | 20 | abbii 2280 | . . . . 5 |
22 | rnopab 4845 | . . . . 5 | |
23 | df-rab 2451 | . . . . 5 | |
24 | 21, 22, 23 | 3eqtr4i 2195 | . . . 4 |
25 | 10, 24 | eqtri 2185 | . . 3 |
26 | 8, 25 | eqtri 2185 | . 2 |
27 | 7, 26 | eqtri 2185 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1342 wex 1479 wcel 2135 cab 2150 crab 2446 copab 4036 cmpt 4037 ccnv 4597 crn 4599 cres 4600 cima 4601 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-mpt 4039 df-xp 4604 df-rel 4605 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 |
This theorem is referenced by: mptiniseg 5092 dmmpt 5093 fmpt 5629 f1oresrab 5644 suppssfv 6040 suppssov1 6041 infrenegsupex 9523 infxrnegsupex 11190 txcnmpt 12820 txdis1cn 12825 imasnopn 12846 |
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