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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 6236 | . 2 tpos tpos |
3 | reldmoprab 5918 | . . 3 | |
4 | dftpos3 6221 | . . 3 tpos | |
5 | 3, 4 | ax-mp 5 | . 2 tpos |
6 | nfcv 2306 | . . . . 5 | |
7 | nfoprab2 5883 | . . . . 5 | |
8 | nfcv 2306 | . . . . 5 | |
9 | 6, 7, 8 | nfbr 4022 | . . . 4 |
10 | nfcv 2306 | . . . . 5 | |
11 | nfoprab1 5882 | . . . . 5 | |
12 | nfcv 2306 | . . . . 5 | |
13 | 10, 11, 12 | nfbr 4022 | . . . 4 |
14 | nfv 1515 | . . . 4 | |
15 | nfv 1515 | . . . 4 | |
16 | opeq12 3754 | . . . . . 6 | |
17 | 16 | ancoms 266 | . . . . 5 |
18 | 17 | breq1d 3986 | . . . 4 |
19 | 9, 13, 14, 15, 18 | cbvoprab12 5907 | . . 3 |
20 | nfcv 2306 | . . . . 5 | |
21 | nfoprab3 5884 | . . . . 5 | |
22 | nfcv 2306 | . . . . 5 | |
23 | 20, 21, 22 | nfbr 4022 | . . . 4 |
24 | nfv 1515 | . . . 4 | |
25 | breq2 3980 | . . . . 5 | |
26 | df-br 3977 | . . . . . 6 | |
27 | oprabid 5865 | . . . . . 6 | |
28 | 26, 27 | bitri 183 | . . . . 5 |
29 | 25, 28 | bitrdi 195 | . . . 4 |
30 | 23, 24, 29 | cbvoprab3 5909 | . . 3 |
31 | 19, 30 | eqtri 2185 | . 2 |
32 | 2, 5, 31 | 3eqtri 2189 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1342 wcel 2135 cop 3573 class class class wbr 3976 cdm 4598 wrel 4603 coprab 5837 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-in1 604 ax-in2 605 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-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 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-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 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-oprab 5840 df-tpos 6204 |
This theorem is referenced by: tposmpo 6240 |
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