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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 5084 | . . . . 5 | |
2 | 1 | breq1d 3986 | . . . 4 |
3 | 2 | 3adant3 1006 | . . 3 |
4 | 3 | anbi2d 460 | . 2 |
5 | brtpos2 6210 | . . 3 tpos | |
6 | 5 | 3ad2ant3 1009 | . 2 tpos |
7 | opexg 4200 | . . . . . . . . 9 | |
8 | 7 | ancoms 266 | . . . . . . . 8 |
9 | 8 | anim1i 338 | . . . . . . 7 |
10 | 9 | 3impa 1183 | . . . . . 6 |
11 | breldmg 4804 | . . . . . . 7 | |
12 | 11 | 3expia 1194 | . . . . . 6 |
13 | 10, 12 | syl 14 | . . . . 5 |
14 | opelcnvg 4778 | . . . . . 6 | |
15 | 14 | 3adant3 1006 | . . . . 5 |
16 | 13, 15 | sylibrd 168 | . . . 4 |
17 | elun1 3284 | . . . 4 | |
18 | 16, 17 | syl6 33 | . . 3 |
19 | 18 | pm4.71rd 392 | . 2 |
20 | 4, 6, 19 | 3bitr4d 219 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wcel 2135 cvv 2721 cun 3109 c0 3404 csn 3570 cop 3573 cuni 3783 class class class wbr 3976 ccnv 4597 cdm 4598 tpos ctpos 6203 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 df-tpos 6204 |
This theorem is referenced by: ottposg 6214 dmtpos 6215 rntpos 6216 ovtposg 6218 dftpos3 6221 tpostpos 6223 |
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