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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 6477 | . . 3 |
4 | 1, 2 | ecopqsi 6477 | . . 3 |
5 | df-br 3925 | . . . . 5 | |
6 | brecop.5 | . . . . . 6 | |
7 | 6 | eleq2i 2204 | . . . . 5 |
8 | 5, 7 | bitri 183 | . . . 4 |
9 | eqeq1 2144 | . . . . . . . 8 | |
10 | 9 | anbi1d 460 | . . . . . . 7 |
11 | 10 | anbi1d 460 | . . . . . 6 |
12 | 11 | 4exbidv 1842 | . . . . 5 |
13 | eqeq1 2144 | . . . . . . . 8 | |
14 | 13 | anbi2d 459 | . . . . . . 7 |
15 | 14 | anbi1d 460 | . . . . . 6 |
16 | 15 | 4exbidv 1842 | . . . . 5 |
17 | 12, 16 | opelopab2 4187 | . . . 4 |
18 | 8, 17 | syl5bb 191 | . . 3 |
19 | 3, 4, 18 | syl2an 287 | . 2 |
20 | opeq12 3702 | . . . . . 6 | |
21 | 20 | eceq1d 6458 | . . . . 5 |
22 | opeq12 3702 | . . . . . 6 | |
23 | 22 | eceq1d 6458 | . . . . 5 |
24 | 21, 23 | anim12i 336 | . . . 4 |
25 | opelxpi 4566 | . . . . . . . 8 | |
26 | opelxp 4564 | . . . . . . . . 9 | |
27 | brecop.2 | . . . . . . . . . . 11 | |
28 | 27 | a1i 9 | . . . . . . . . . 10 |
29 | id 19 | . . . . . . . . . 10 | |
30 | 28, 29 | ereldm 6465 | . . . . . . . . 9 |
31 | 26, 30 | syl5bbr 193 | . . . . . . . 8 |
32 | 25, 31 | syl5ibr 155 | . . . . . . 7 |
33 | opelxpi 4566 | . . . . . . . 8 | |
34 | opelxp 4564 | . . . . . . . . 9 | |
35 | 27 | a1i 9 | . . . . . . . . . 10 |
36 | id 19 | . . . . . . . . . 10 | |
37 | 35, 36 | ereldm 6465 | . . . . . . . . 9 |
38 | 34, 37 | syl5bbr 193 | . . . . . . . 8 |
39 | 33, 38 | syl5ibr 155 | . . . . . . 7 |
40 | 32, 39 | im2anan9 587 | . . . . . 6 |
41 | brecop.6 | . . . . . . . . 9 | |
42 | 41 | an4s 577 | . . . . . . . 8 |
43 | 42 | ex 114 | . . . . . . 7 |
44 | 43 | com13 80 | . . . . . 6 |
45 | 40, 44 | mpdd 41 | . . . . 5 |
46 | 45 | pm5.74d 181 | . . . 4 |
47 | 24, 46 | cgsex4g 2718 | . . 3 |
48 | eqcom 2139 | . . . . . . 7 | |
49 | eqcom 2139 | . . . . . . 7 | |
50 | 48, 49 | anbi12i 455 | . . . . . 6 |
51 | 50 | a1i 9 | . . . . 5 |
52 | biimt 240 | . . . . 5 | |
53 | 51, 52 | anbi12d 464 | . . . 4 |
54 | 53 | 4exbidv 1842 | . . 3 |
55 | biimt 240 | . . 3 | |
56 | 47, 54, 55 | 3bitr4d 219 | . 2 |
57 | 19, 56 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cvv 2681 cop 3525 class class class wbr 3924 copab 3983 cxp 4532 wer 6419 cec 6420 cqs 6421 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-er 6422 df-ec 6424 df-qs 6428 |
This theorem is referenced by: ordpipqqs 7175 ltsrprg 7548 |
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