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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 5684 | . . . 4 | |
2 | 1 | ad2ant2rl 502 | . . 3 |
3 | fcofo 5685 | . . . . 5 | |
4 | 3 | 3expa 1181 | . . . 4 |
5 | 4 | adantrr 470 | . . 3 |
6 | df-f1o 5130 | . . 3 | |
7 | 2, 5, 6 | sylanbrc 413 | . 2 |
8 | simprl 520 | . . . 4 | |
9 | 8 | coeq2d 4701 | . . 3 |
10 | coass 5057 | . . . 4 | |
11 | f1ococnv1 5396 | . . . . . . 7 | |
12 | 7, 11 | syl 14 | . . . . . 6 |
13 | 12 | coeq1d 4700 | . . . . 5 |
14 | fcoi2 5304 | . . . . . 6 | |
15 | 14 | ad2antlr 480 | . . . . 5 |
16 | 13, 15 | eqtrd 2172 | . . . 4 |
17 | 10, 16 | syl5eqr 2186 | . . 3 |
18 | f1ocnv 5380 | . . . 4 | |
19 | f1of 5367 | . . . 4 | |
20 | fcoi1 5303 | . . . 4 | |
21 | 7, 18, 19, 20 | 4syl 18 | . . 3 |
22 | 9, 17, 21 | 3eqtr3rd 2181 | . 2 |
23 | 7, 22 | jca 304 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 cid 4210 ccnv 4538 cres 4541 ccom 4543 wf 5119 wf1 5120 wfo 5121 wf1o 5122 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 |
This theorem is referenced by: txswaphmeo 12490 |
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