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Mirrors > Home > ILE Home > Th. List > pofun | Unicode version |
Description: A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.) |
Ref | Expression |
---|---|
pofun.1 | |
pofun.2 |
Ref | Expression |
---|---|
pofun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcsb1v 2882 | . . . . . . 7 | |
2 | 1 | nfel1 2188 | . . . . . 6 |
3 | csbeq1a 2860 | . . . . . . 7 | |
4 | 3 | eleq1d 2106 | . . . . . 6 |
5 | 2, 4 | rspc 2650 | . . . . 5 |
6 | 5 | impcom 116 | . . . 4 |
7 | poirr 4044 | . . . . 5 | |
8 | df-br 3765 | . . . . . 6 | |
9 | pofun.1 | . . . . . . 7 | |
10 | 9 | eleq2i 2104 | . . . . . 6 |
11 | nfcv 2178 | . . . . . . . 8 | |
12 | nfcv 2178 | . . . . . . . 8 | |
13 | 1, 11, 12 | nfbr 3808 | . . . . . . 7 |
14 | nfv 1421 | . . . . . . 7 | |
15 | vex 2560 | . . . . . . 7 | |
16 | 3 | breq1d 3774 | . . . . . . 7 |
17 | vex 2560 | . . . . . . . . . 10 | |
18 | pofun.2 | . . . . . . . . . 10 | |
19 | 17, 12, 18 | csbief 2891 | . . . . . . . . 9 |
20 | csbeq1 2855 | . . . . . . . . 9 | |
21 | 19, 20 | syl5eqr 2086 | . . . . . . . 8 |
22 | 21 | breq2d 3776 | . . . . . . 7 |
23 | 13, 14, 15, 15, 16, 22 | opelopabf 4011 | . . . . . 6 |
24 | 8, 10, 23 | 3bitri 195 | . . . . 5 |
25 | 7, 24 | sylnibr 602 | . . . 4 |
26 | 6, 25 | sylan2 270 | . . 3 |
27 | 26 | anassrs 380 | . 2 |
28 | 5 | com12 27 | . . . . . 6 |
29 | nfcsb1v 2882 | . . . . . . . . 9 | |
30 | 29 | nfel1 2188 | . . . . . . . 8 |
31 | csbeq1a 2860 | . . . . . . . . 9 | |
32 | 31 | eleq1d 2106 | . . . . . . . 8 |
33 | 30, 32 | rspc 2650 | . . . . . . 7 |
34 | 33 | com12 27 | . . . . . 6 |
35 | nfcsb1v 2882 | . . . . . . . . 9 | |
36 | 35 | nfel1 2188 | . . . . . . . 8 |
37 | csbeq1a 2860 | . . . . . . . . 9 | |
38 | 37 | eleq1d 2106 | . . . . . . . 8 |
39 | 36, 38 | rspc 2650 | . . . . . . 7 |
40 | 39 | com12 27 | . . . . . 6 |
41 | 28, 34, 40 | 3anim123d 1214 | . . . . 5 |
42 | 41 | imp 115 | . . . 4 |
43 | 42 | adantll 445 | . . 3 |
44 | potr 4045 | . . . . 5 | |
45 | df-br 3765 | . . . . . . 7 | |
46 | 9 | eleq2i 2104 | . . . . . . 7 |
47 | nfv 1421 | . . . . . . . 8 | |
48 | vex 2560 | . . . . . . . 8 | |
49 | csbeq1 2855 | . . . . . . . . . 10 | |
50 | 19, 49 | syl5eqr 2086 | . . . . . . . . 9 |
51 | 50 | breq2d 3776 | . . . . . . . 8 |
52 | 13, 47, 15, 48, 16, 51 | opelopabf 4011 | . . . . . . 7 |
53 | 45, 46, 52 | 3bitri 195 | . . . . . 6 |
54 | df-br 3765 | . . . . . . 7 | |
55 | 9 | eleq2i 2104 | . . . . . . 7 |
56 | 29, 11, 12 | nfbr 3808 | . . . . . . . 8 |
57 | nfv 1421 | . . . . . . . 8 | |
58 | vex 2560 | . . . . . . . 8 | |
59 | 31 | breq1d 3774 | . . . . . . . 8 |
60 | csbeq1 2855 | . . . . . . . . . 10 | |
61 | 19, 60 | syl5eqr 2086 | . . . . . . . . 9 |
62 | 61 | breq2d 3776 | . . . . . . . 8 |
63 | 56, 57, 48, 58, 59, 62 | opelopabf 4011 | . . . . . . 7 |
64 | 54, 55, 63 | 3bitri 195 | . . . . . 6 |
65 | 53, 64 | anbi12i 433 | . . . . 5 |
66 | df-br 3765 | . . . . . 6 | |
67 | 9 | eleq2i 2104 | . . . . . 6 |
68 | nfv 1421 | . . . . . . 7 | |
69 | 61 | breq2d 3776 | . . . . . . 7 |
70 | 13, 68, 15, 58, 16, 69 | opelopabf 4011 | . . . . . 6 |
71 | 66, 67, 70 | 3bitri 195 | . . . . 5 |
72 | 44, 65, 71 | 3imtr4g 194 | . . . 4 |
73 | 72 | adantlr 446 | . . 3 |
74 | 43, 73 | syldan 266 | . 2 |
75 | 27, 74 | ispod 4041 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 wral 2306 csb 2852 cop 3378 class class class wbr 3764 copab 3817 wpo 4031 |
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-in1 544 ax-in2 545 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-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 |
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-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-po 4033 |
This theorem is referenced by: (None) |
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