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Mirrors > Home > ILE Home > Th. List > brecop | Unicode version |
Description: Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.) |
Ref | Expression |
---|---|
brecop.1 | |
brecop.2 | |
brecop.4 | |
brecop.5 | |
brecop.6 |
Ref | Expression |
---|---|
brecop |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brecop.1 | . . . 4 | |
2 | brecop.4 | . . . 4 | |
3 | 1, 2 | ecopqsi 6161 | . . 3 |
4 | 1, 2 | ecopqsi 6161 | . . 3 |
5 | df-br 3765 | . . . . 5 | |
6 | brecop.5 | . . . . . 6 | |
7 | 6 | eleq2i 2104 | . . . . 5 |
8 | 5, 7 | bitri 173 | . . . 4 |
9 | eqeq1 2046 | . . . . . . . 8 | |
10 | 9 | anbi1d 438 | . . . . . . 7 |
11 | 10 | anbi1d 438 | . . . . . 6 |
12 | 11 | 4exbidv 1750 | . . . . 5 |
13 | eqeq1 2046 | . . . . . . . 8 | |
14 | 13 | anbi2d 437 | . . . . . . 7 |
15 | 14 | anbi1d 438 | . . . . . 6 |
16 | 15 | 4exbidv 1750 | . . . . 5 |
17 | 12, 16 | opelopab2 4007 | . . . 4 |
18 | 8, 17 | syl5bb 181 | . . 3 |
19 | 3, 4, 18 | syl2an 273 | . 2 |
20 | opeq12 3551 | . . . . . 6 | |
21 | 20 | eceq1d 6142 | . . . . 5 |
22 | opeq12 3551 | . . . . . 6 | |
23 | 22 | eceq1d 6142 | . . . . 5 |
24 | 21, 23 | anim12i 321 | . . . 4 |
25 | opelxpi 4376 | . . . . . . . 8 | |
26 | opelxp 4374 | . . . . . . . . 9 | |
27 | brecop.2 | . . . . . . . . . . 11 | |
28 | 27 | a1i 9 | . . . . . . . . . 10 |
29 | id 19 | . . . . . . . . . 10 | |
30 | 28, 29 | ereldm 6149 | . . . . . . . . 9 |
31 | 26, 30 | syl5bbr 183 | . . . . . . . 8 |
32 | 25, 31 | syl5ibr 145 | . . . . . . 7 |
33 | opelxpi 4376 | . . . . . . . 8 | |
34 | opelxp 4374 | . . . . . . . . 9 | |
35 | 27 | a1i 9 | . . . . . . . . . 10 |
36 | id 19 | . . . . . . . . . 10 | |
37 | 35, 36 | ereldm 6149 | . . . . . . . . 9 |
38 | 34, 37 | syl5bbr 183 | . . . . . . . 8 |
39 | 33, 38 | syl5ibr 145 | . . . . . . 7 |
40 | 32, 39 | im2anan9 530 | . . . . . 6 |
41 | brecop.6 | . . . . . . . . 9 | |
42 | 41 | an4s 522 | . . . . . . . 8 |
43 | 42 | ex 108 | . . . . . . 7 |
44 | 43 | com13 74 | . . . . . 6 |
45 | 40, 44 | mpdd 36 | . . . . 5 |
46 | 45 | pm5.74d 171 | . . . 4 |
47 | 24, 46 | cgsex4g 2591 | . . 3 |
48 | eqcom 2042 | . . . . . . 7 | |
49 | eqcom 2042 | . . . . . . 7 | |
50 | 48, 49 | anbi12i 433 | . . . . . 6 |
51 | 50 | a1i 9 | . . . . 5 |
52 | biimt 230 | . . . . 5 | |
53 | 51, 52 | anbi12d 442 | . . . 4 |
54 | 53 | 4exbidv 1750 | . . 3 |
55 | biimt 230 | . . 3 | |
56 | 47, 54, 55 | 3bitr4d 209 | . 2 |
57 | 19, 56 | bitrd 177 | 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 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-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 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 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-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 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-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-er 6106 df-ec 6108 df-qs 6112 |
This theorem is referenced by: ordpipqqs 6472 ltsrprg 6832 |
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