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Mirrors > Home > ILE Home > Th. List > mpt22eqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5628. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
mpt22eqb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm13.183 2732 | . . . . . 6 | |
2 | 1 | ralimi 2426 | . . . . 5 |
3 | ralbi 2489 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | 4 | ralimi 2426 | . . 3 |
6 | ralbi 2489 | . . 3 | |
7 | 5, 6 | syl 14 | . 2 |
8 | df-mpt2 5537 | . . . 4 | |
9 | df-mpt2 5537 | . . . 4 | |
10 | 8, 9 | eqeq12i 2094 | . . 3 |
11 | eqoprab2b 5583 | . . 3 | |
12 | pm5.32 440 | . . . . . . 7 | |
13 | 12 | albii 1399 | . . . . . 6 |
14 | 19.21v 1794 | . . . . . 6 | |
15 | 13, 14 | bitr3i 184 | . . . . 5 |
16 | 15 | 2albii 1400 | . . . 4 |
17 | r2al 2385 | . . . 4 | |
18 | 16, 17 | bitr4i 185 | . . 3 |
19 | 10, 11, 18 | 3bitri 204 | . 2 |
20 | 7, 19 | syl6rbbr 197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wal 1282 wceq 1284 wcel 1433 wral 2348 coprab 5533 cmpt2 5534 |
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-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-oprab 5536 df-mpt2 5537 |
This theorem is referenced by: (None) |
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