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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcof1 5805 |
. . . 4
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2 | 1 | ad2ant2rl 511 |
. . 3
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3 | fcofo 5806 |
. . . . 5
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4 | 3 | 3expa 1205 |
. . . 4
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5 | 4 | adantrr 479 |
. . 3
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6 | df-f1o 5242 |
. . 3
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7 | 2, 5, 6 | sylanbrc 417 |
. 2
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8 | simprl 529 |
. . . 4
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9 | 8 | coeq2d 4807 |
. . 3
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10 | coass 5165 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | f1ococnv1 5509 |
. . . . . . 7
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12 | 7, 11 | syl 14 |
. . . . . 6
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13 | 12 | coeq1d 4806 |
. . . . 5
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14 | fcoi2 5416 |
. . . . . 6
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15 | 14 | ad2antlr 489 |
. . . . 5
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16 | 13, 15 | eqtrd 2222 |
. . . 4
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17 | 10, 16 | eqtr3id 2236 |
. . 3
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18 | f1ocnv 5493 |
. . . 4
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19 | f1of 5480 |
. . . 4
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20 | fcoi1 5415 |
. . . 4
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21 | 7, 18, 19, 20 | 4syl 18 |
. . 3
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22 | 9, 17, 21 | 3eqtr3rd 2231 |
. 2
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23 | 7, 22 | jca 306 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 |
This theorem is referenced by: txswaphmeo 14298 |
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