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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 3035 | . . . . . . 7 | |
2 | 1 | nfel1 2292 | . . . . . 6 |
3 | csbeq1a 3012 | . . . . . . 7 | |
4 | 3 | eleq1d 2208 | . . . . . 6 |
5 | 2, 4 | rspc 2783 | . . . . 5 |
6 | 5 | impcom 124 | . . . 4 |
7 | poirr 4229 | . . . . 5 | |
8 | df-br 3930 | . . . . . 6 | |
9 | pofun.1 | . . . . . . 7 | |
10 | 9 | eleq2i 2206 | . . . . . 6 |
11 | nfcv 2281 | . . . . . . . 8 | |
12 | nfcv 2281 | . . . . . . . 8 | |
13 | 1, 11, 12 | nfbr 3974 | . . . . . . 7 |
14 | nfv 1508 | . . . . . . 7 | |
15 | vex 2689 | . . . . . . 7 | |
16 | 3 | breq1d 3939 | . . . . . . 7 |
17 | vex 2689 | . . . . . . . . . 10 | |
18 | pofun.2 | . . . . . . . . . 10 | |
19 | 17, 12, 18 | csbief 3044 | . . . . . . . . 9 |
20 | csbeq1 3006 | . . . . . . . . 9 | |
21 | 19, 20 | syl5eqr 2186 | . . . . . . . 8 |
22 | 21 | breq2d 3941 | . . . . . . 7 |
23 | 13, 14, 15, 15, 16, 22 | opelopabf 4196 | . . . . . 6 |
24 | 8, 10, 23 | 3bitri 205 | . . . . 5 |
25 | 7, 24 | sylnibr 666 | . . . 4 |
26 | 6, 25 | sylan2 284 | . . 3 |
27 | 26 | anassrs 397 | . 2 |
28 | 5 | com12 30 | . . . . . 6 |
29 | nfcsb1v 3035 | . . . . . . . . 9 | |
30 | 29 | nfel1 2292 | . . . . . . . 8 |
31 | csbeq1a 3012 | . . . . . . . . 9 | |
32 | 31 | eleq1d 2208 | . . . . . . . 8 |
33 | 30, 32 | rspc 2783 | . . . . . . 7 |
34 | 33 | com12 30 | . . . . . 6 |
35 | nfcsb1v 3035 | . . . . . . . . 9 | |
36 | 35 | nfel1 2292 | . . . . . . . 8 |
37 | csbeq1a 3012 | . . . . . . . . 9 | |
38 | 37 | eleq1d 2208 | . . . . . . . 8 |
39 | 36, 38 | rspc 2783 | . . . . . . 7 |
40 | 39 | com12 30 | . . . . . 6 |
41 | 28, 34, 40 | 3anim123d 1297 | . . . . 5 |
42 | 41 | imp 123 | . . . 4 |
43 | 42 | adantll 467 | . . 3 |
44 | potr 4230 | . . . . 5 | |
45 | df-br 3930 | . . . . . . 7 | |
46 | 9 | eleq2i 2206 | . . . . . . 7 |
47 | nfv 1508 | . . . . . . . 8 | |
48 | vex 2689 | . . . . . . . 8 | |
49 | csbeq1 3006 | . . . . . . . . . 10 | |
50 | 19, 49 | syl5eqr 2186 | . . . . . . . . 9 |
51 | 50 | breq2d 3941 | . . . . . . . 8 |
52 | 13, 47, 15, 48, 16, 51 | opelopabf 4196 | . . . . . . 7 |
53 | 45, 46, 52 | 3bitri 205 | . . . . . 6 |
54 | df-br 3930 | . . . . . . 7 | |
55 | 9 | eleq2i 2206 | . . . . . . 7 |
56 | 29, 11, 12 | nfbr 3974 | . . . . . . . 8 |
57 | nfv 1508 | . . . . . . . 8 | |
58 | vex 2689 | . . . . . . . 8 | |
59 | 31 | breq1d 3939 | . . . . . . . 8 |
60 | csbeq1 3006 | . . . . . . . . . 10 | |
61 | 19, 60 | syl5eqr 2186 | . . . . . . . . 9 |
62 | 61 | breq2d 3941 | . . . . . . . 8 |
63 | 56, 57, 48, 58, 59, 62 | opelopabf 4196 | . . . . . . 7 |
64 | 54, 55, 63 | 3bitri 205 | . . . . . 6 |
65 | 53, 64 | anbi12i 455 | . . . . 5 |
66 | df-br 3930 | . . . . . 6 | |
67 | 9 | eleq2i 2206 | . . . . . 6 |
68 | nfv 1508 | . . . . . . 7 | |
69 | 61 | breq2d 3941 | . . . . . . 7 |
70 | 13, 68, 15, 58, 16, 69 | opelopabf 4196 | . . . . . 6 |
71 | 66, 67, 70 | 3bitri 205 | . . . . 5 |
72 | 44, 65, 71 | 3imtr4g 204 | . . . 4 |
73 | 72 | adantlr 468 | . . 3 |
74 | 43, 73 | syldan 280 | . 2 |
75 | 27, 74 | ispod 4226 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wral 2416 csb 3003 cop 3530 class class class wbr 3929 copab 3988 wpo 4216 |
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-in1 603 ax-in2 604 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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-po 4218 |
This theorem is referenced by: (None) |
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