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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 2938 | . . . . . . 7 | |
2 | 1 | nfel1 2229 | . . . . . 6 |
3 | csbeq1a 2916 | . . . . . . 7 | |
4 | 3 | eleq1d 2147 | . . . . . 6 |
5 | 2, 4 | rspc 2695 | . . . . 5 |
6 | 5 | impcom 123 | . . . 4 |
7 | poirr 4062 | . . . . 5 | |
8 | df-br 3786 | . . . . . 6 | |
9 | pofun.1 | . . . . . . 7 | |
10 | 9 | eleq2i 2145 | . . . . . 6 |
11 | nfcv 2219 | . . . . . . . 8 | |
12 | nfcv 2219 | . . . . . . . 8 | |
13 | 1, 11, 12 | nfbr 3829 | . . . . . . 7 |
14 | nfv 1461 | . . . . . . 7 | |
15 | vex 2604 | . . . . . . 7 | |
16 | 3 | breq1d 3795 | . . . . . . 7 |
17 | vex 2604 | . . . . . . . . . 10 | |
18 | pofun.2 | . . . . . . . . . 10 | |
19 | 17, 12, 18 | csbief 2947 | . . . . . . . . 9 |
20 | csbeq1 2911 | . . . . . . . . 9 | |
21 | 19, 20 | syl5eqr 2127 | . . . . . . . 8 |
22 | 21 | breq2d 3797 | . . . . . . 7 |
23 | 13, 14, 15, 15, 16, 22 | opelopabf 4029 | . . . . . 6 |
24 | 8, 10, 23 | 3bitri 204 | . . . . 5 |
25 | 7, 24 | sylnibr 634 | . . . 4 |
26 | 6, 25 | sylan2 280 | . . 3 |
27 | 26 | anassrs 392 | . 2 |
28 | 5 | com12 30 | . . . . . 6 |
29 | nfcsb1v 2938 | . . . . . . . . 9 | |
30 | 29 | nfel1 2229 | . . . . . . . 8 |
31 | csbeq1a 2916 | . . . . . . . . 9 | |
32 | 31 | eleq1d 2147 | . . . . . . . 8 |
33 | 30, 32 | rspc 2695 | . . . . . . 7 |
34 | 33 | com12 30 | . . . . . 6 |
35 | nfcsb1v 2938 | . . . . . . . . 9 | |
36 | 35 | nfel1 2229 | . . . . . . . 8 |
37 | csbeq1a 2916 | . . . . . . . . 9 | |
38 | 37 | eleq1d 2147 | . . . . . . . 8 |
39 | 36, 38 | rspc 2695 | . . . . . . 7 |
40 | 39 | com12 30 | . . . . . 6 |
41 | 28, 34, 40 | 3anim123d 1250 | . . . . 5 |
42 | 41 | imp 122 | . . . 4 |
43 | 42 | adantll 459 | . . 3 |
44 | potr 4063 | . . . . 5 | |
45 | df-br 3786 | . . . . . . 7 | |
46 | 9 | eleq2i 2145 | . . . . . . 7 |
47 | nfv 1461 | . . . . . . . 8 | |
48 | vex 2604 | . . . . . . . 8 | |
49 | csbeq1 2911 | . . . . . . . . . 10 | |
50 | 19, 49 | syl5eqr 2127 | . . . . . . . . 9 |
51 | 50 | breq2d 3797 | . . . . . . . 8 |
52 | 13, 47, 15, 48, 16, 51 | opelopabf 4029 | . . . . . . 7 |
53 | 45, 46, 52 | 3bitri 204 | . . . . . 6 |
54 | df-br 3786 | . . . . . . 7 | |
55 | 9 | eleq2i 2145 | . . . . . . 7 |
56 | 29, 11, 12 | nfbr 3829 | . . . . . . . 8 |
57 | nfv 1461 | . . . . . . . 8 | |
58 | vex 2604 | . . . . . . . 8 | |
59 | 31 | breq1d 3795 | . . . . . . . 8 |
60 | csbeq1 2911 | . . . . . . . . . 10 | |
61 | 19, 60 | syl5eqr 2127 | . . . . . . . . 9 |
62 | 61 | breq2d 3797 | . . . . . . . 8 |
63 | 56, 57, 48, 58, 59, 62 | opelopabf 4029 | . . . . . . 7 |
64 | 54, 55, 63 | 3bitri 204 | . . . . . 6 |
65 | 53, 64 | anbi12i 447 | . . . . 5 |
66 | df-br 3786 | . . . . . 6 | |
67 | 9 | eleq2i 2145 | . . . . . 6 |
68 | nfv 1461 | . . . . . . 7 | |
69 | 61 | breq2d 3797 | . . . . . . 7 |
70 | 13, 68, 15, 58, 16, 69 | opelopabf 4029 | . . . . . 6 |
71 | 66, 67, 70 | 3bitri 204 | . . . . 5 |
72 | 44, 65, 71 | 3imtr4g 203 | . . . 4 |
73 | 72 | adantlr 460 | . . 3 |
74 | 43, 73 | syldan 276 | . 2 |
75 | 27, 74 | ispod 4059 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 wral 2348 csb 2908 cop 3401 class class class wbr 3785 copab 3838 wpo 4049 |
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-in1 576 ax-in2 577 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-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 |
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-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-po 4051 |
This theorem is referenced by: (None) |
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