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Mirrors > Home > ILE Home > Th. List > brtposg | Unicode version |
Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.) |
Ref | Expression |
---|---|
brtposg | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opswapg 4827 | . . . . 5 | |
2 | 1 | breq1d 3795 | . . . 4 |
3 | 2 | 3adant3 958 | . . 3 |
4 | 3 | anbi2d 451 | . 2 |
5 | brtpos2 5889 | . . 3 tpos | |
6 | 5 | 3ad2ant3 961 | . 2 tpos |
7 | opexg 3983 | . . . . . . . . 9 | |
8 | 7 | ancoms 264 | . . . . . . . 8 |
9 | 8 | anim1i 333 | . . . . . . 7 |
10 | 9 | 3impa 1133 | . . . . . 6 |
11 | breldmg 4559 | . . . . . . 7 | |
12 | 11 | 3expia 1140 | . . . . . 6 |
13 | 10, 12 | syl 14 | . . . . 5 |
14 | opelcnvg 4533 | . . . . . 6 | |
15 | 14 | 3adant3 958 | . . . . 5 |
16 | 13, 15 | sylibrd 167 | . . . 4 |
17 | elun1 3139 | . . . 4 | |
18 | 16, 17 | syl6 33 | . . 3 |
19 | 18 | pm4.71rd 386 | . 2 |
20 | 4, 6, 19 | 3bitr4d 218 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wcel 1433 cvv 2601 cun 2971 c0 3251 csn 3398 cop 3401 cuni 3601 class class class wbr 3785 ccnv 4362 cdm 4363 tpos ctpos 5882 |
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-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-13 1444 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 ax-un 4188 |
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-rab 2357 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 df-tpos 5883 |
This theorem is referenced by: ottposg 5893 dmtpos 5894 rntpos 5895 ovtposg 5897 dftpos3 5900 tpostpos 5902 |
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