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Mirrors > Home > ILE Home > Th. List > mpt22eqb | Unicode version |
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5608. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
mpt22eqb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm13.183 2681 | . . . . . 6 | |
2 | 1 | ralimi 2384 | . . . . 5 |
3 | ralbi 2445 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | 4 | ralimi 2384 | . . 3 |
6 | ralbi 2445 | . . 3 | |
7 | 5, 6 | syl 14 | . 2 |
8 | df-mpt2 5517 | . . . 4 | |
9 | df-mpt2 5517 | . . . 4 | |
10 | 8, 9 | eqeq12i 2053 | . . 3 |
11 | eqoprab2b 5563 | . . 3 | |
12 | pm5.32 426 | . . . . . . 7 | |
13 | 12 | albii 1359 | . . . . . 6 |
14 | 19.21v 1753 | . . . . . 6 | |
15 | 13, 14 | bitr3i 175 | . . . . 5 |
16 | 15 | 2albii 1360 | . . . 4 |
17 | r2al 2343 | . . . 4 | |
18 | 16, 17 | bitr4i 176 | . . 3 |
19 | 10, 11, 18 | 3bitri 195 | . 2 |
20 | 7, 19 | syl6rbbr 188 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wal 1241 wceq 1243 wcel 1393 wral 2306 coprab 5513 cmpt2 5514 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-oprab 5516 df-mpt2 5517 |
This theorem is referenced by: (None) |
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