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Mirrors > Home > ILE Home > Th. List > tposoprab | Unicode version |
Description: Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposoprab.1 |
Ref | Expression |
---|---|
tposoprab | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tposoprab.1 | . . 3 | |
2 | 1 | tposeqi 6142 | . 2 tpos tpos |
3 | reldmoprab 5824 | . . 3 | |
4 | dftpos3 6127 | . . 3 tpos | |
5 | 3, 4 | ax-mp 5 | . 2 tpos |
6 | nfcv 2258 | . . . . 5 | |
7 | nfoprab2 5789 | . . . . 5 | |
8 | nfcv 2258 | . . . . 5 | |
9 | 6, 7, 8 | nfbr 3944 | . . . 4 |
10 | nfcv 2258 | . . . . 5 | |
11 | nfoprab1 5788 | . . . . 5 | |
12 | nfcv 2258 | . . . . 5 | |
13 | 10, 11, 12 | nfbr 3944 | . . . 4 |
14 | nfv 1493 | . . . 4 | |
15 | nfv 1493 | . . . 4 | |
16 | opeq12 3677 | . . . . . 6 | |
17 | 16 | ancoms 266 | . . . . 5 |
18 | 17 | breq1d 3909 | . . . 4 |
19 | 9, 13, 14, 15, 18 | cbvoprab12 5813 | . . 3 |
20 | nfcv 2258 | . . . . 5 | |
21 | nfoprab3 5790 | . . . . 5 | |
22 | nfcv 2258 | . . . . 5 | |
23 | 20, 21, 22 | nfbr 3944 | . . . 4 |
24 | nfv 1493 | . . . 4 | |
25 | breq2 3903 | . . . . 5 | |
26 | df-br 3900 | . . . . . 6 | |
27 | oprabid 5771 | . . . . . 6 | |
28 | 26, 27 | bitri 183 | . . . . 5 |
29 | 25, 28 | syl6bb 195 | . . . 4 |
30 | 23, 24, 29 | cbvoprab3 5815 | . . 3 |
31 | 19, 30 | eqtri 2138 | . 2 |
32 | 2, 5, 31 | 3eqtri 2142 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1316 wcel 1465 cop 3500 class class class wbr 3899 cdm 4509 wrel 4514 coprab 5743 tpos ctpos 6109 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 df-oprab 5746 df-tpos 6110 |
This theorem is referenced by: tposmpo 6146 |
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