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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 2140 |
. 2
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2 | snexg 4116 |
. . . 4
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3 | cnvexg 5084 |
. . . 4
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4 | uniexg 4369 |
. . . 4
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5 | 2, 3, 4 | 3syl 17 |
. . 3
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6 | 5 | adantl 275 |
. 2
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7 | snexg 4116 |
. . . 4
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8 | cnvexg 5084 |
. . . 4
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9 | uniexg 4369 |
. . . 4
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10 | 7, 8, 9 | 3syl 17 |
. . 3
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11 | 10 | adantl 275 |
. 2
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12 | cnvf1olem 6129 |
. . 3
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13 | relcnv 4925 |
. . . . 5
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14 | simpr 109 |
. . . . 5
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15 | cnvf1olem 6129 |
. . . . 5
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16 | 13, 14, 15 | sylancr 411 |
. . . 4
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17 | dfrel2 4997 |
. . . . . . 7
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18 | eleq2 2204 |
. . . . . . 7
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19 | 17, 18 | sylbi 120 |
. . . . . 6
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20 | 19 | anbi1d 461 |
. . . . 5
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21 | 20 | adantr 274 |
. . . 4
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22 | 16, 21 | mpbid 146 |
. . 3
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23 | 12, 22 | impbida 586 |
. 2
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24 | 1, 6, 11, 23 | f1od 5981 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1st 6046 df-2nd 6047 |
This theorem is referenced by: tposf12 6174 cnven 6710 xpcomf1o 6727 fsumcnv 11238 |
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