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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 3884 | . . . . 5 | |
2 | 1 | eldm 4532 | . . . 4 |
3 | vex 2560 | . . . . . . 7 | |
4 | brtpos0 5867 | . . . . . . 7 tpos | |
5 | 3, 4 | ax-mp 7 | . . . . . 6 tpos |
6 | 0nelxp 4372 | . . . . . . . 8 | |
7 | df-rel 4352 | . . . . . . . . 9 tpos tpos | |
8 | ssel 2939 | . . . . . . . . 9 tpos tpos | |
9 | 7, 8 | sylbi 114 | . . . . . . . 8 tpos tpos |
10 | 6, 9 | mtoi 590 | . . . . . . 7 tpos tpos |
11 | 1, 3 | breldm 4539 | . . . . . . 7 tpos tpos |
12 | 10, 11 | nsyl3 556 | . . . . . 6 tpos tpos |
13 | 5, 12 | sylbir 125 | . . . . 5 tpos |
14 | 13 | exlimiv 1489 | . . . 4 tpos |
15 | 2, 14 | sylbi 114 | . . 3 tpos |
16 | 15 | con2i 557 | . 2 tpos |
17 | vex 2560 | . . . . . 6 | |
18 | 17 | eldm 4532 | . . . . 5 tpos tpos |
19 | relcnv 4703 | . . . . . . . . . . 11 | |
20 | df-rel 4352 | . . . . . . . . . . 11 | |
21 | 19, 20 | mpbi 133 | . . . . . . . . . 10 |
22 | 21 | sseli 2941 | . . . . . . . . 9 |
23 | 22 | a1i 9 | . . . . . . . 8 tpos |
24 | elsni 3393 | . . . . . . . . . . . 12 | |
25 | 24 | breq1d 3774 | . . . . . . . . . . 11 tpos tpos |
26 | 1, 3 | breldm 4539 | . . . . . . . . . . . . 13 |
27 | 26 | pm2.24d 552 | . . . . . . . . . . . 12 |
28 | 5, 27 | sylbi 114 | . . . . . . . . . . 11 tpos |
29 | 25, 28 | syl6bi 152 | . . . . . . . . . 10 tpos |
30 | 29 | com3l 75 | . . . . . . . . 9 tpos |
31 | 30 | impcom 116 | . . . . . . . 8 tpos |
32 | brtpos2 5866 | . . . . . . . . . . . 12 tpos | |
33 | 3, 32 | ax-mp 7 | . . . . . . . . . . 11 tpos |
34 | 33 | simplbi 259 | . . . . . . . . . 10 tpos |
35 | elun 3084 | . . . . . . . . . 10 | |
36 | 34, 35 | sylib 127 | . . . . . . . . 9 tpos |
37 | 36 | adantl 262 | . . . . . . . 8 tpos |
38 | 23, 31, 37 | mpjaod 638 | . . . . . . 7 tpos |
39 | 38 | ex 108 | . . . . . 6 tpos |
40 | 39 | exlimdv 1700 | . . . . 5 tpos |
41 | 18, 40 | syl5bi 141 | . . . 4 tpos |
42 | 41 | ssrdv 2951 | . . 3 tpos |
43 | 42, 7 | sylibr 137 | . 2 tpos |
44 | 16, 43 | impbii 117 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 wex 1381 wcel 1393 cvv 2557 cun 2915 wss 2917 c0 3224 csn 3375 cuni 3580 class class class wbr 3764 cxp 4343 ccnv 4344 cdm 4345 wrel 4350 tpos ctpos 5859 |
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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 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-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 df-tpos 5860 |
This theorem is referenced by: dmtpos 5871 |
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