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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 5025 | . . . . 5 | |
2 | 1 | breq1d 3939 | . . . 4 |
3 | 2 | 3adant3 1001 | . . 3 |
4 | 3 | anbi2d 459 | . 2 |
5 | brtpos2 6148 | . . 3 tpos | |
6 | 5 | 3ad2ant3 1004 | . 2 tpos |
7 | opexg 4150 | . . . . . . . . 9 | |
8 | 7 | ancoms 266 | . . . . . . . 8 |
9 | 8 | anim1i 338 | . . . . . . 7 |
10 | 9 | 3impa 1176 | . . . . . 6 |
11 | breldmg 4745 | . . . . . . 7 | |
12 | 11 | 3expia 1183 | . . . . . 6 |
13 | 10, 12 | syl 14 | . . . . 5 |
14 | opelcnvg 4719 | . . . . . 6 | |
15 | 14 | 3adant3 1001 | . . . . 5 |
16 | 13, 15 | sylibrd 168 | . . . 4 |
17 | elun1 3243 | . . . 4 | |
18 | 16, 17 | syl6 33 | . . 3 |
19 | 18 | pm4.71rd 391 | . 2 |
20 | 4, 6, 19 | 3bitr4d 219 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wcel 1480 cvv 2686 cun 3069 c0 3363 csn 3527 cop 3530 cuni 3736 class class class wbr 3929 ccnv 4538 cdm 4539 tpos ctpos 6141 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 df-tpos 6142 |
This theorem is referenced by: ottposg 6152 dmtpos 6153 rntpos 6154 ovtposg 6156 dftpos3 6159 tpostpos 6161 |
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