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Mirrors > Home > ILE Home > Th. List > cnvf1o | Unicode version |
Description: Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Ref | Expression |
---|---|
cnvf1o |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2088 |
. 2
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2 | snexg 4019 |
. . . 4
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3 | cnvexg 4968 |
. . . 4
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4 | uniexg 4265 |
. . . 4
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5 | 2, 3, 4 | 3syl 17 |
. . 3
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6 | 5 | adantl 271 |
. 2
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7 | snexg 4019 |
. . . 4
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8 | cnvexg 4968 |
. . . 4
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9 | uniexg 4265 |
. . . 4
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10 | 7, 8, 9 | 3syl 17 |
. . 3
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11 | 10 | adantl 271 |
. 2
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12 | cnvf1olem 5989 |
. . 3
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13 | relcnv 4810 |
. . . . 5
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14 | simpr 108 |
. . . . 5
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15 | cnvf1olem 5989 |
. . . . 5
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16 | 13, 14, 15 | sylancr 405 |
. . . 4
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17 | dfrel2 4881 |
. . . . . . 7
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18 | eleq2 2151 |
. . . . . . 7
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19 | 17, 18 | sylbi 119 |
. . . . . 6
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20 | 19 | anbi1d 453 |
. . . . 5
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21 | 20 | adantr 270 |
. . . 4
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22 | 16, 21 | mpbid 145 |
. . 3
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23 | 12, 22 | impbida 563 |
. 2
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24 | 1, 6, 11, 23 | f1od 5847 |
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 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-sbc 2841 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-1st 5911 df-2nd 5912 |
This theorem is referenced by: tposf12 6034 cnven 6523 xpcomf1o 6539 fsumcnv 10827 |
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