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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 5576 |
. . . 4
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2 | 1 | ad2ant2rl 496 |
. . 3
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3 | fcofo 5577 |
. . . . 5
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4 | 3 | 3expa 1144 |
. . . 4
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5 | 4 | adantrr 464 |
. . 3
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6 | df-f1o 5035 |
. . 3
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7 | 2, 5, 6 | sylanbrc 409 |
. 2
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8 | simprl 499 |
. . . 4
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9 | 8 | coeq2d 4611 |
. . 3
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10 | coass 4962 |
. . . 4
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11 | f1ococnv1 5295 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 7, 11 | syl 14 |
. . . . . 6
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13 | 12 | coeq1d 4610 |
. . . . 5
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14 | fcoi2 5205 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 14 | ad2antlr 474 |
. . . . 5
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16 | 13, 15 | eqtrd 2121 |
. . . 4
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17 | 10, 16 | syl5eqr 2135 |
. . 3
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18 | f1ocnv 5279 |
. . . 4
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19 | f1of 5266 |
. . . 4
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20 | fcoi1 5204 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 7, 18, 19, 20 | 4syl 18 |
. . 3
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22 | 9, 17, 21 | 3eqtr3rd 2130 |
. 2
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23 | 7, 22 | jca 301 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-sbc 2842 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 |
This theorem is referenced by: (None) |
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