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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 6256 | . 2 tpos tpos |
3 | reldmoprab 5938 | . . 3 | |
4 | dftpos3 6241 | . . 3 tpos | |
5 | 3, 4 | ax-mp 5 | . 2 tpos |
6 | nfcv 2312 | . . . . 5 | |
7 | nfoprab2 5903 | . . . . 5 | |
8 | nfcv 2312 | . . . . 5 | |
9 | 6, 7, 8 | nfbr 4035 | . . . 4 |
10 | nfcv 2312 | . . . . 5 | |
11 | nfoprab1 5902 | . . . . 5 | |
12 | nfcv 2312 | . . . . 5 | |
13 | 10, 11, 12 | nfbr 4035 | . . . 4 |
14 | nfv 1521 | . . . 4 | |
15 | nfv 1521 | . . . 4 | |
16 | opeq12 3767 | . . . . . 6 | |
17 | 16 | ancoms 266 | . . . . 5 |
18 | 17 | breq1d 3999 | . . . 4 |
19 | 9, 13, 14, 15, 18 | cbvoprab12 5927 | . . 3 |
20 | nfcv 2312 | . . . . 5 | |
21 | nfoprab3 5904 | . . . . 5 | |
22 | nfcv 2312 | . . . . 5 | |
23 | 20, 21, 22 | nfbr 4035 | . . . 4 |
24 | nfv 1521 | . . . 4 | |
25 | breq2 3993 | . . . . 5 | |
26 | df-br 3990 | . . . . . 6 | |
27 | oprabid 5885 | . . . . . 6 | |
28 | 26, 27 | bitri 183 | . . . . 5 |
29 | 25, 28 | bitrdi 195 | . . . 4 |
30 | 23, 24, 29 | cbvoprab3 5929 | . . 3 |
31 | 19, 30 | eqtri 2191 | . 2 |
32 | 2, 5, 31 | 3eqtri 2195 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1348 wcel 2141 cop 3586 class class class wbr 3989 cdm 4611 wrel 4616 coprab 5854 tpos ctpos 6223 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-fv 5206 df-oprab 5857 df-tpos 6224 |
This theorem is referenced by: tposmpo 6260 |
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