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Mirrors > Home > ILE Home > Th. List > f1ococnv1 | Unicode version |
Description: The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.) |
Ref | Expression |
---|---|
f1ococnv1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1orel 5495 |
. . . 4
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2 | dfrel2 5108 |
. . . 4
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3 | 1, 2 | sylib 122 |
. . 3
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4 | 3 | coeq2d 4818 |
. 2
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5 | f1ocnv 5505 |
. . 3
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6 | f1ococnv2 5519 |
. . 3
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7 | 5, 6 | syl 14 |
. 2
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8 | 4, 7 | eqtr3d 2228 |
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 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-br 4030 df-opab 4091 df-id 4322 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 |
This theorem is referenced by: f1cocnv1 5522 f1ocnvfv1 5812 fcof1o 5824 mapen 6893 hashfacen 10897 |
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