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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 4690 | . . . . 5 |
7 | oviec.9 | . . . . . 6 | |
8 | 7, 5 | opbrop 4690 | . . . . 5 |
9 | 6, 8 | bi2anan9 601 | . . . 4 |
10 | oviec.2 | . . . . . . 7 | |
11 | oviec.11 | . . . . . . 7 | |
12 | oviec.10 | . . . . . . 7 | |
13 | 10, 11, 12 | ovi3 5989 | . . . . . 6 |
14 | oviec.3 | . . . . . . 7 | |
15 | oviec.12 | . . . . . . 7 | |
16 | 14, 15, 12 | ovi3 5989 | . . . . . 6 |
17 | 13, 16 | breqan12d 4005 | . . . . 5 |
18 | 17 | an4s 583 | . . . 4 |
19 | 3, 9, 18 | 3imtr4d 202 | . . 3 |
20 | oviec.14 | . . . 4 | |
21 | oviec.15 | . . . . . . . 8 | |
22 | 21 | eleq2i 2237 | . . . . . . 7 |
23 | 21 | eleq2i 2237 | . . . . . . 7 |
24 | 22, 23 | anbi12i 457 | . . . . . 6 |
25 | 24 | anbi1i 455 | . . . . 5 |
26 | 25 | oprabbii 5908 | . . . 4 |
27 | 20, 26 | eqtri 2191 | . . 3 |
28 | 1, 2, 19, 27 | th3q 6618 | . 2 |
29 | oviec.1 | . . . 4 | |
30 | oviec.13 | . . . 4 | |
31 | 29, 30, 12 | ovi3 5989 | . . 3 |
32 | 31 | eceq1d 6549 | . 2 |
33 | 28, 32 | eqtrd 2203 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 cvv 2730 cop 3586 class class class wbr 3989 copab 4049 cxp 4609 (class class class)co 5853 coprab 5854 wer 6510 cec 6511 cqs 6512 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-er 6513 df-ec 6515 df-qs 6519 |
This theorem is referenced by: addpipqqs 7332 mulpipqqs 7335 |
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