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Mirrors > Home > MPE Home > Th. List > brecop | Structured version Visualization version 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 7804 | . . 3 |
4 | 1, 2 | ecopqsi 7804 | . . 3 |
5 | df-br 4654 | . . . . 5 | |
6 | brecop.5 | . . . . . 6 | |
7 | 6 | eleq2i 2693 | . . . . 5 |
8 | 5, 7 | bitri 264 | . . . 4 |
9 | eqeq1 2626 | . . . . . . . 8 | |
10 | 9 | anbi1d 741 | . . . . . . 7 |
11 | 10 | anbi1d 741 | . . . . . 6 |
12 | 11 | 4exbidv 1854 | . . . . 5 |
13 | eqeq1 2626 | . . . . . . . 8 | |
14 | 13 | anbi2d 740 | . . . . . . 7 |
15 | 14 | anbi1d 741 | . . . . . 6 |
16 | 15 | 4exbidv 1854 | . . . . 5 |
17 | 12, 16 | opelopab2 4996 | . . . 4 |
18 | 8, 17 | syl5bb 272 | . . 3 |
19 | 3, 4, 18 | syl2an 494 | . 2 |
20 | opeq12 4404 | . . . . . 6 | |
21 | 20 | eceq1d 7783 | . . . . 5 |
22 | opeq12 4404 | . . . . . 6 | |
23 | 22 | eceq1d 7783 | . . . . 5 |
24 | 21, 23 | anim12i 590 | . . . 4 |
25 | opelxpi 5148 | . . . . . . . 8 | |
26 | opelxp 5146 | . . . . . . . . 9 | |
27 | brecop.2 | . . . . . . . . . . 11 | |
28 | 27 | a1i 11 | . . . . . . . . . 10 |
29 | id 22 | . . . . . . . . . 10 | |
30 | 28, 29 | ereldm 7790 | . . . . . . . . 9 |
31 | 26, 30 | syl5bbr 274 | . . . . . . . 8 |
32 | 25, 31 | syl5ibr 236 | . . . . . . 7 |
33 | opelxpi 5148 | . . . . . . . 8 | |
34 | opelxp 5146 | . . . . . . . . 9 | |
35 | 27 | a1i 11 | . . . . . . . . . 10 |
36 | id 22 | . . . . . . . . . 10 | |
37 | 35, 36 | ereldm 7790 | . . . . . . . . 9 |
38 | 34, 37 | syl5bbr 274 | . . . . . . . 8 |
39 | 33, 38 | syl5ibr 236 | . . . . . . 7 |
40 | 32, 39 | im2anan9 880 | . . . . . 6 |
41 | brecop.6 | . . . . . . . . 9 | |
42 | 41 | an4s 869 | . . . . . . . 8 |
43 | 42 | ex 450 | . . . . . . 7 |
44 | 43 | com13 88 | . . . . . 6 |
45 | 40, 44 | mpdd 43 | . . . . 5 |
46 | 45 | pm5.74d 262 | . . . 4 |
47 | 24, 46 | cgsex4g 3240 | . . 3 |
48 | eqcom 2629 | . . . . . . 7 | |
49 | eqcom 2629 | . . . . . . 7 | |
50 | 48, 49 | anbi12i 733 | . . . . . 6 |
51 | 50 | a1i 11 | . . . . 5 |
52 | biimt 350 | . . . . 5 | |
53 | 51, 52 | anbi12d 747 | . . . 4 |
54 | 53 | 4exbidv 1854 | . . 3 |
55 | biimt 350 | . . 3 | |
56 | 47, 54, 55 | 3bitr4d 300 | . 2 |
57 | 19, 56 | bitrd 268 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 cvv 3200 cop 4183 class class class wbr 4653 copab 4712 cxp 5112 wer 7739 cec 7740 cqs 7741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-er 7742 df-ec 7744 df-qs 7748 |
This theorem is referenced by: ltsrpr 9898 |
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