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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 2434 | . . . 4 |
3 | f1od.1 | . . . . 5 | |
4 | 3 | fnmpt 5045 | . . . 4 |
5 | 2, 4 | syl 14 | . . 3 |
6 | f1od.3 | . . . . . 6 | |
7 | 6 | ralrimiva 2434 | . . . . 5 |
8 | eqid 2081 | . . . . . 6 | |
9 | 8 | fnmpt 5045 | . . . . 5 |
10 | 7, 9 | syl 14 | . . . 4 |
11 | f1od.4 | . . . . . . 7 | |
12 | 11 | opabbidv 3844 | . . . . . 6 |
13 | df-mpt 3841 | . . . . . . . . 9 | |
14 | 3, 13 | eqtri 2101 | . . . . . . . 8 |
15 | 14 | cnveqi 4528 | . . . . . . 7 |
16 | cnvopab 4746 | . . . . . . 7 | |
17 | 15, 16 | eqtri 2101 | . . . . . 6 |
18 | df-mpt 3841 | . . . . . 6 | |
19 | 12, 17, 18 | 3eqtr4g 2138 | . . . . 5 |
20 | 19 | fneq1d 5009 | . . . 4 |
21 | 10, 20 | mpbird 165 | . . 3 |
22 | dff1o4 5154 | . . 3 | |
23 | 5, 21, 22 | sylanbrc 408 | . 2 |
24 | 23, 19 | jca 300 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 copab 3838 cmpt 3839 ccnv 4362 wfn 4917 wf1o 4921 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 |
This theorem is referenced by: f1od 5723 f1ocnv2d 5724 |
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