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Mirrors > Home > ILE Home > Th. List > reldmtpos | Unicode version |
Description: Necessary and sufficient condition for tpos to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
reldmtpos | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4025 | . . . . 5 | |
2 | 1 | eldm 4706 | . . . 4 |
3 | vex 2663 | . . . . . . 7 | |
4 | brtpos0 6117 | . . . . . . 7 tpos | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 tpos |
6 | 0nelxp 4537 | . . . . . . . 8 | |
7 | df-rel 4516 | . . . . . . . . 9 tpos tpos | |
8 | ssel 3061 | . . . . . . . . 9 tpos tpos | |
9 | 7, 8 | sylbi 120 | . . . . . . . 8 tpos tpos |
10 | 6, 9 | mtoi 638 | . . . . . . 7 tpos tpos |
11 | 1, 3 | breldm 4713 | . . . . . . 7 tpos tpos |
12 | 10, 11 | nsyl3 600 | . . . . . 6 tpos tpos |
13 | 5, 12 | sylbir 134 | . . . . 5 tpos |
14 | 13 | exlimiv 1562 | . . . 4 tpos |
15 | 2, 14 | sylbi 120 | . . 3 tpos |
16 | 15 | con2i 601 | . 2 tpos |
17 | vex 2663 | . . . . . 6 | |
18 | 17 | eldm 4706 | . . . . 5 tpos tpos |
19 | relcnv 4887 | . . . . . . . . . . 11 | |
20 | df-rel 4516 | . . . . . . . . . . 11 | |
21 | 19, 20 | mpbi 144 | . . . . . . . . . 10 |
22 | 21 | sseli 3063 | . . . . . . . . 9 |
23 | 22 | a1i 9 | . . . . . . . 8 tpos |
24 | elsni 3515 | . . . . . . . . . . . 12 | |
25 | 24 | breq1d 3909 | . . . . . . . . . . 11 tpos tpos |
26 | 1, 3 | breldm 4713 | . . . . . . . . . . . . 13 |
27 | 26 | pm2.24d 596 | . . . . . . . . . . . 12 |
28 | 5, 27 | sylbi 120 | . . . . . . . . . . 11 tpos |
29 | 25, 28 | syl6bi 162 | . . . . . . . . . 10 tpos |
30 | 29 | com3l 81 | . . . . . . . . 9 tpos |
31 | 30 | impcom 124 | . . . . . . . 8 tpos |
32 | brtpos2 6116 | . . . . . . . . . . . 12 tpos | |
33 | 3, 32 | ax-mp 5 | . . . . . . . . . . 11 tpos |
34 | 33 | simplbi 272 | . . . . . . . . . 10 tpos |
35 | elun 3187 | . . . . . . . . . 10 | |
36 | 34, 35 | sylib 121 | . . . . . . . . 9 tpos |
37 | 36 | adantl 275 | . . . . . . . 8 tpos |
38 | 23, 31, 37 | mpjaod 692 | . . . . . . 7 tpos |
39 | 38 | ex 114 | . . . . . 6 tpos |
40 | 39 | exlimdv 1775 | . . . . 5 tpos |
41 | 18, 40 | syl5bi 151 | . . . 4 tpos |
42 | 41 | ssrdv 3073 | . . 3 tpos |
43 | 42, 7 | sylibr 133 | . 2 tpos |
44 | 16, 43 | impbii 125 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 wex 1453 wcel 1465 cvv 2660 cun 3039 wss 3041 c0 3333 csn 3497 cuni 3706 class class class wbr 3899 cxp 4507 ccnv 4508 cdm 4509 wrel 4514 tpos ctpos 6109 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 df-tpos 6110 |
This theorem is referenced by: dmtpos 6121 |
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