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Mirrors > Home > ILE Home > Th. List > f1orescnv | Unicode version |
Description: The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.) |
Ref | Expression |
---|---|
f1orescnv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 5281 |
. . 3
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2 | 1 | adantl 272 |
. 2
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3 | funcnvres 5102 |
. . . 4
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4 | df-ima 4467 |
. . . . . 6
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5 | dff1o5 5277 |
. . . . . . 7
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6 | 5 | simprbi 270 |
. . . . . 6
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7 | 4, 6 | syl5eq 2133 |
. . . . 5
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8 | 7 | reseq2d 4728 |
. . . 4
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9 | 3, 8 | sylan9eq 2141 |
. . 3
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10 | f1oeq1 5259 |
. . 3
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11 | 9, 10 | syl 14 |
. 2
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12 | 2, 11 | mpbid 146 |
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 3965 ax-pow 4017 ax-pr 4047 |
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 2624 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-br 3854 df-opab 3908 df-id 4131 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 |
This theorem is referenced by: f1oresrab 5479 |
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