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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 5652 | . . . 4 | |
2 | 1 | ad2ant2rl 502 | . . 3 |
3 | fcofo 5653 | . . . . 5 | |
4 | 3 | 3expa 1166 | . . . 4 |
5 | 4 | adantrr 470 | . . 3 |
6 | df-f1o 5100 | . . 3 | |
7 | 2, 5, 6 | sylanbrc 413 | . 2 |
8 | simprl 505 | . . . 4 | |
9 | 8 | coeq2d 4671 | . . 3 |
10 | coass 5027 | . . . 4 | |
11 | f1ococnv1 5364 | . . . . . . 7 | |
12 | 7, 11 | syl 14 | . . . . . 6 |
13 | 12 | coeq1d 4670 | . . . . 5 |
14 | fcoi2 5274 | . . . . . 6 | |
15 | 14 | ad2antlr 480 | . . . . 5 |
16 | 13, 15 | eqtrd 2150 | . . . 4 |
17 | 10, 16 | syl5eqr 2164 | . . 3 |
18 | f1ocnv 5348 | . . . 4 | |
19 | f1of 5335 | . . . 4 | |
20 | fcoi1 5273 | . . . 4 | |
21 | 7, 18, 19, 20 | 4syl 18 | . . 3 |
22 | 9, 17, 21 | 3eqtr3rd 2159 | . 2 |
23 | 7, 22 | jca 304 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 cid 4180 ccnv 4508 cres 4511 ccom 4513 wf 5089 wf1 5090 wfo 5091 wf1o 5092 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 |
This theorem is referenced by: txswaphmeo 12417 |
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