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Mirrors > Home > ILE Home > Th. List > f1ococnv1 | Unicode version |
Description: The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.) |
Ref | Expression |
---|---|
f1ococnv1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1orel 5149 | . . . 4 | |
2 | dfrel2 4791 | . . . 4 | |
3 | 1, 2 | sylib 120 | . . 3 |
4 | 3 | coeq2d 4516 | . 2 |
5 | f1ocnv 5159 | . . 3 | |
6 | f1ococnv2 5173 | . . 3 | |
7 | 5, 6 | syl 14 | . 2 |
8 | 4, 7 | eqtr3d 2115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1284 cid 4043 ccnv 4362 cres 4365 ccom 4367 wrel 4368 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-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 |
This theorem is referenced by: f1cocnv1 5176 f1ocnvfv1 5437 fcof1o 5449 |
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