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Mirrors > Home > ILE Home > Th. List > mpo2eqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 6005. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
mpo2eqb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpo 5902 |
. . . 4
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2 | df-mpo 5902 |
. . . 4
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3 | 1, 2 | eqeq12i 2203 |
. . 3
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4 | eqoprab2b 5955 |
. . 3
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5 | pm5.32 453 |
. . . . . . 7
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6 | 5 | albii 1481 |
. . . . . 6
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7 | 19.21v 1884 |
. . . . . 6
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8 | 6, 7 | bitr3i 186 |
. . . . 5
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9 | 8 | 2albii 1482 |
. . . 4
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10 | r2al 2509 |
. . . 4
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11 | 9, 10 | bitr4i 187 |
. . 3
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12 | 3, 4, 11 | 3bitri 206 |
. 2
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13 | pm13.183 2890 |
. . . . . 6
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14 | 13 | ralimi 2553 |
. . . . 5
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15 | ralbi 2622 |
. . . . 5
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16 | 14, 15 | syl 14 |
. . . 4
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17 | 16 | ralimi 2553 |
. . 3
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18 | ralbi 2622 |
. . 3
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19 | 17, 18 | syl 14 |
. 2
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20 | 12, 19 | bitr4id 199 |
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-in1 615 ax-in2 616 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 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-oprab 5901 df-mpo 5902 |
This theorem is referenced by: (None) |
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