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Mirrors > Home > ILE Home > Th. List > f1ocnvd | Unicode version |
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
f1od.1 | |
f1od.2 | |
f1od.3 | |
f1od.4 |
Ref | Expression |
---|---|
f1ocnvd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1od.2 | . . . . 5 | |
2 | 1 | ralrimiva 2392 | . . . 4 |
3 | f1od.1 | . . . . 5 | |
4 | 3 | fnmpt 5025 | . . . 4 |
5 | 2, 4 | syl 14 | . . 3 |
6 | f1od.3 | . . . . . 6 | |
7 | 6 | ralrimiva 2392 | . . . . 5 |
8 | eqid 2040 | . . . . . 6 | |
9 | 8 | fnmpt 5025 | . . . . 5 |
10 | 7, 9 | syl 14 | . . . 4 |
11 | f1od.4 | . . . . . . 7 | |
12 | 11 | opabbidv 3823 | . . . . . 6 |
13 | df-mpt 3820 | . . . . . . . . 9 | |
14 | 3, 13 | eqtri 2060 | . . . . . . . 8 |
15 | 14 | cnveqi 4510 | . . . . . . 7 |
16 | cnvopab 4726 | . . . . . . 7 | |
17 | 15, 16 | eqtri 2060 | . . . . . 6 |
18 | df-mpt 3820 | . . . . . 6 | |
19 | 12, 17, 18 | 3eqtr4g 2097 | . . . . 5 |
20 | 19 | fneq1d 4989 | . . . 4 |
21 | 10, 20 | mpbird 156 | . . 3 |
22 | dff1o4 5134 | . . 3 | |
23 | 5, 21, 22 | sylanbrc 394 | . 2 |
24 | 23, 19 | jca 290 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 copab 3817 cmpt 3818 ccnv 4344 wfn 4897 wf1o 4901 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: f1od 5703 f1ocnv2d 5704 |
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