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Mirrors > Home > ILE Home > Th. List > oviec | Unicode version |
Description: Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.) |
Ref | Expression |
---|---|
oviec.1 | |
oviec.2 | |
oviec.3 | |
oviec.4 | |
oviec.5 | |
oviec.7 | |
oviec.8 | |
oviec.9 | |
oviec.10 | |
oviec.11 | |
oviec.12 | |
oviec.13 | |
oviec.14 | |
oviec.15 | |
oviec.16 |
Ref | Expression |
---|---|
oviec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oviec.4 | . . 3 | |
2 | oviec.5 | . . 3 | |
3 | oviec.16 | . . . 4 | |
4 | oviec.8 | . . . . . 6 | |
5 | oviec.7 | . . . . . 6 | |
6 | 4, 5 | opbrop 4419 | . . . . 5 |
7 | oviec.9 | . . . . . 6 | |
8 | 7, 5 | opbrop 4419 | . . . . 5 |
9 | 6, 8 | bi2anan9 538 | . . . 4 |
10 | oviec.2 | . . . . . . 7 | |
11 | oviec.11 | . . . . . . 7 | |
12 | oviec.10 | . . . . . . 7 | |
13 | 10, 11, 12 | ovi3 5637 | . . . . . 6 |
14 | oviec.3 | . . . . . . 7 | |
15 | oviec.12 | . . . . . . 7 | |
16 | 14, 15, 12 | ovi3 5637 | . . . . . 6 |
17 | 13, 16 | breqan12d 3779 | . . . . 5 |
18 | 17 | an4s 522 | . . . 4 |
19 | 3, 9, 18 | 3imtr4d 192 | . . 3 |
20 | oviec.14 | . . . 4 | |
21 | oviec.15 | . . . . . . . 8 | |
22 | 21 | eleq2i 2104 | . . . . . . 7 |
23 | 21 | eleq2i 2104 | . . . . . . 7 |
24 | 22, 23 | anbi12i 433 | . . . . . 6 |
25 | 24 | anbi1i 431 | . . . . 5 |
26 | 25 | oprabbii 5560 | . . . 4 |
27 | 20, 26 | eqtri 2060 | . . 3 |
28 | 1, 2, 19, 27 | th3q 6211 | . 2 |
29 | oviec.1 | . . . 4 | |
30 | oviec.13 | . . . 4 | |
31 | 29, 30, 12 | ovi3 5637 | . . 3 |
32 | 31 | eceq1d 6142 | . 2 |
33 | 28, 32 | eqtrd 2072 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 cop 3378 class class class wbr 3764 copab 3817 cxp 4343 (class class class)co 5512 coprab 5513 wer 6103 cec 6104 cqs 6105 |
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-13 1404 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-un 4170 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-rex 2312 df-v 2559 df-sbc 2765 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-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fv 4910 df-ov 5515 df-oprab 5516 df-er 6106 df-ec 6108 df-qs 6112 |
This theorem is referenced by: addpipqqs 6468 mulpipqqs 6471 |
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