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Mirrors > Home > ILE Home > Th. List > fcof1o | Unicode version |
Description: Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
fcof1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcof1 5423 | . . . 4 | |
2 | 1 | ad2ant2rl 480 | . . 3 |
3 | fcofo 5424 | . . . . 5 | |
4 | 3 | 3expa 1104 | . . . 4 |
5 | 4 | adantrr 448 | . . 3 |
6 | df-f1o 4909 | . . 3 | |
7 | 2, 5, 6 | sylanbrc 394 | . 2 |
8 | simprl 483 | . . . 4 | |
9 | 8 | coeq2d 4498 | . . 3 |
10 | coass 4839 | . . . 4 | |
11 | f1ococnv1 5155 | . . . . . . 7 | |
12 | 7, 11 | syl 14 | . . . . . 6 |
13 | 12 | coeq1d 4497 | . . . . 5 |
14 | fcoi2 5071 | . . . . . 6 | |
15 | 14 | ad2antlr 458 | . . . . 5 |
16 | 13, 15 | eqtrd 2072 | . . . 4 |
17 | 10, 16 | syl5eqr 2086 | . . 3 |
18 | f1ocnv 5139 | . . . 4 | |
19 | f1of 5126 | . . . 4 | |
20 | fcoi1 5070 | . . . 4 | |
21 | 7, 18, 19, 20 | 4syl 18 | . . 3 |
22 | 9, 17, 21 | 3eqtr3rd 2081 | . 2 |
23 | 7, 22 | jca 290 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 cid 4025 ccnv 4344 cres 4347 ccom 4349 wf 4898 wf1 4899 wfo 4900 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-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 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-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 |
This theorem is referenced by: (None) |
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