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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 5020 | . . . . 5 | |
2 | 1 | breq1d 3934 | . . . 4 |
3 | 2 | 3adant3 1001 | . . 3 |
4 | 3 | anbi2d 459 | . 2 |
5 | brtpos2 6141 | . . 3 tpos | |
6 | 5 | 3ad2ant3 1004 | . 2 tpos |
7 | opexg 4145 | . . . . . . . . 9 | |
8 | 7 | ancoms 266 | . . . . . . . 8 |
9 | 8 | anim1i 338 | . . . . . . 7 |
10 | 9 | 3impa 1176 | . . . . . 6 |
11 | breldmg 4740 | . . . . . . 7 | |
12 | 11 | 3expia 1183 | . . . . . 6 |
13 | 10, 12 | syl 14 | . . . . 5 |
14 | opelcnvg 4714 | . . . . . 6 | |
15 | 14 | 3adant3 1001 | . . . . 5 |
16 | 13, 15 | sylibrd 168 | . . . 4 |
17 | elun1 3238 | . . . 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 2681 cun 3064 c0 3358 csn 3522 cop 3525 cuni 3731 class class class wbr 3924 ccnv 4533 cdm 4534 tpos ctpos 6134 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 df-tpos 6135 |
This theorem is referenced by: ottposg 6145 dmtpos 6146 rntpos 6147 ovtposg 6149 dftpos3 6152 tpostpos 6154 |
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