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Mirrors > Home > ILE Home > Th. List > mpo2eqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5830. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
mpo2eqb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm13.183 2790 |
. . . . . 6
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2 | 1 | ralimi 2467 |
. . . . 5
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3 | ralbi 2536 |
. . . . 5
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4 | 2, 3 | syl 14 |
. . . 4
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5 | 4 | ralimi 2467 |
. . 3
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6 | ralbi 2536 |
. . 3
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7 | 5, 6 | syl 14 |
. 2
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8 | df-mpo 5731 |
. . . 4
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9 | df-mpo 5731 |
. . . 4
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10 | 8, 9 | eqeq12i 2126 |
. . 3
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11 | eqoprab2b 5781 |
. . 3
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12 | pm5.32 446 |
. . . . . . 7
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13 | 12 | albii 1427 |
. . . . . 6
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14 | 19.21v 1825 |
. . . . . 6
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15 | 13, 14 | bitr3i 185 |
. . . . 5
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16 | 15 | 2albii 1428 |
. . . 4
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17 | r2al 2426 |
. . . 4
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18 | 16, 17 | bitr4i 186 |
. . 3
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19 | 10, 11, 18 | 3bitri 205 |
. 2
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20 | 7, 19 | syl6rbbr 198 |
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-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-setind 4410 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-oprab 5730 df-mpo 5731 |
This theorem is referenced by: (None) |
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