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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 6184 | . . 3 |
4 | 1, 2 | ecopqsi 6184 | . . 3 |
5 | df-br 3786 | . . . . 5 | |
6 | brecop.5 | . . . . . 6 | |
7 | 6 | eleq2i 2145 | . . . . 5 |
8 | 5, 7 | bitri 182 | . . . 4 |
9 | eqeq1 2087 | . . . . . . . 8 | |
10 | 9 | anbi1d 452 | . . . . . . 7 |
11 | 10 | anbi1d 452 | . . . . . 6 |
12 | 11 | 4exbidv 1791 | . . . . 5 |
13 | eqeq1 2087 | . . . . . . . 8 | |
14 | 13 | anbi2d 451 | . . . . . . 7 |
15 | 14 | anbi1d 452 | . . . . . 6 |
16 | 15 | 4exbidv 1791 | . . . . 5 |
17 | 12, 16 | opelopab2 4025 | . . . 4 |
18 | 8, 17 | syl5bb 190 | . . 3 |
19 | 3, 4, 18 | syl2an 283 | . 2 |
20 | opeq12 3572 | . . . . . 6 | |
21 | 20 | eceq1d 6165 | . . . . 5 |
22 | opeq12 3572 | . . . . . 6 | |
23 | 22 | eceq1d 6165 | . . . . 5 |
24 | 21, 23 | anim12i 331 | . . . 4 |
25 | opelxpi 4394 | . . . . . . . 8 | |
26 | opelxp 4392 | . . . . . . . . 9 | |
27 | brecop.2 | . . . . . . . . . . 11 | |
28 | 27 | a1i 9 | . . . . . . . . . 10 |
29 | id 19 | . . . . . . . . . 10 | |
30 | 28, 29 | ereldm 6172 | . . . . . . . . 9 |
31 | 26, 30 | syl5bbr 192 | . . . . . . . 8 |
32 | 25, 31 | syl5ibr 154 | . . . . . . 7 |
33 | opelxpi 4394 | . . . . . . . 8 | |
34 | opelxp 4392 | . . . . . . . . 9 | |
35 | 27 | a1i 9 | . . . . . . . . . 10 |
36 | id 19 | . . . . . . . . . 10 | |
37 | 35, 36 | ereldm 6172 | . . . . . . . . 9 |
38 | 34, 37 | syl5bbr 192 | . . . . . . . 8 |
39 | 33, 38 | syl5ibr 154 | . . . . . . 7 |
40 | 32, 39 | im2anan9 562 | . . . . . 6 |
41 | brecop.6 | . . . . . . . . 9 | |
42 | 41 | an4s 552 | . . . . . . . 8 |
43 | 42 | ex 113 | . . . . . . 7 |
44 | 43 | com13 79 | . . . . . 6 |
45 | 40, 44 | mpdd 40 | . . . . 5 |
46 | 45 | pm5.74d 180 | . . . 4 |
47 | 24, 46 | cgsex4g 2636 | . . 3 |
48 | eqcom 2083 | . . . . . . 7 | |
49 | eqcom 2083 | . . . . . . 7 | |
50 | 48, 49 | anbi12i 447 | . . . . . 6 |
51 | 50 | a1i 9 | . . . . 5 |
52 | biimt 239 | . . . . 5 | |
53 | 51, 52 | anbi12d 456 | . . . 4 |
54 | 53 | 4exbidv 1791 | . . 3 |
55 | biimt 239 | . . 3 | |
56 | 47, 54, 55 | 3bitr4d 218 | . 2 |
57 | 19, 56 | bitrd 186 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 cvv 2601 cop 3401 class class class wbr 3785 copab 3838 cxp 4361 wer 6126 cec 6127 cqs 6128 |
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-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-13 1444 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-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 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-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-er 6129 df-ec 6131 df-qs 6135 |
This theorem is referenced by: ordpipqqs 6564 ltsrprg 6924 |
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