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Mirrors > Home > ILE Home > Th. List > mpo2eqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5878. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
mpo2eqb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm13.183 2822 | . . . . . 6 | |
2 | 1 | ralimi 2495 | . . . . 5 |
3 | ralbi 2564 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | 4 | ralimi 2495 | . . 3 |
6 | ralbi 2564 | . . 3 | |
7 | 5, 6 | syl 14 | . 2 |
8 | df-mpo 5779 | . . . 4 | |
9 | df-mpo 5779 | . . . 4 | |
10 | 8, 9 | eqeq12i 2153 | . . 3 |
11 | eqoprab2b 5829 | . . 3 | |
12 | pm5.32 448 | . . . . . . 7 | |
13 | 12 | albii 1446 | . . . . . 6 |
14 | 19.21v 1845 | . . . . . 6 | |
15 | 13, 14 | bitr3i 185 | . . . . 5 |
16 | 15 | 2albii 1447 | . . . 4 |
17 | r2al 2454 | . . . 4 | |
18 | 16, 17 | bitr4i 186 | . . 3 |
19 | 10, 11, 18 | 3bitri 205 | . 2 |
20 | 7, 19 | syl6rbbr 198 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wceq 1331 wcel 1480 wral 2416 coprab 5775 cmpo 5776 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-oprab 5778 df-mpo 5779 |
This theorem is referenced by: (None) |
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