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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 5443 | . . . 4 | |
2 | 1 | ad2ant2rl 494 | . . 3 |
3 | fcofo 5444 | . . . . 5 | |
4 | 3 | 3expa 1138 | . . . 4 |
5 | 4 | adantrr 462 | . . 3 |
6 | df-f1o 4929 | . . 3 | |
7 | 2, 5, 6 | sylanbrc 408 | . 2 |
8 | simprl 497 | . . . 4 | |
9 | 8 | coeq2d 4516 | . . 3 |
10 | coass 4859 | . . . 4 | |
11 | f1ococnv1 5175 | . . . . . . 7 | |
12 | 7, 11 | syl 14 | . . . . . 6 |
13 | 12 | coeq1d 4515 | . . . . 5 |
14 | fcoi2 5091 | . . . . . 6 | |
15 | 14 | ad2antlr 472 | . . . . 5 |
16 | 13, 15 | eqtrd 2113 | . . . 4 |
17 | 10, 16 | syl5eqr 2127 | . . 3 |
18 | f1ocnv 5159 | . . . 4 | |
19 | f1of 5146 | . . . 4 | |
20 | fcoi1 5090 | . . . 4 | |
21 | 7, 18, 19, 20 | 4syl 18 | . . 3 |
22 | 9, 17, 21 | 3eqtr3rd 2122 | . 2 |
23 | 7, 22 | jca 300 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 cid 4043 ccnv 4362 cres 4365 ccom 4367 wf 4918 wf1 4919 wfo 4920 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-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 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-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 |
This theorem is referenced by: (None) |
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