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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 2503 | . . . 4 |
3 | f1od.1 | . . . . 5 | |
4 | 3 | fnmpt 5244 | . . . 4 |
5 | 2, 4 | syl 14 | . . 3 |
6 | f1od.3 | . . . . . 6 | |
7 | 6 | ralrimiva 2503 | . . . . 5 |
8 | eqid 2137 | . . . . . 6 | |
9 | 8 | fnmpt 5244 | . . . . 5 |
10 | 7, 9 | syl 14 | . . . 4 |
11 | f1od.4 | . . . . . . 7 | |
12 | 11 | opabbidv 3989 | . . . . . 6 |
13 | df-mpt 3986 | . . . . . . . . 9 | |
14 | 3, 13 | eqtri 2158 | . . . . . . . 8 |
15 | 14 | cnveqi 4709 | . . . . . . 7 |
16 | cnvopab 4935 | . . . . . . 7 | |
17 | 15, 16 | eqtri 2158 | . . . . . 6 |
18 | df-mpt 3986 | . . . . . 6 | |
19 | 12, 17, 18 | 3eqtr4g 2195 | . . . . 5 |
20 | 19 | fneq1d 5208 | . . . 4 |
21 | 10, 20 | mpbird 166 | . . 3 |
22 | dff1o4 5368 | . . 3 | |
23 | 5, 21, 22 | sylanbrc 413 | . 2 |
24 | 23, 19 | jca 304 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 copab 3983 cmpt 3984 ccnv 4533 wfn 5113 wf1o 5117 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 |
This theorem is referenced by: f1od 5966 f1ocnv2d 5967 |
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