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Mirrors > Home > ILE Home > Th. List > cnvcnv | Unicode version |
Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) |
Ref | Expression |
---|---|
cnvcnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4887 | . . . . 5 | |
2 | df-rel 4516 | . . . . 5 | |
3 | 1, 2 | mpbi 144 | . . . 4 |
4 | relxp 4618 | . . . . 5 | |
5 | dfrel2 4959 | . . . . 5 | |
6 | 4, 5 | mpbi 144 | . . . 4 |
7 | 3, 6 | sseqtrri 3102 | . . 3 |
8 | dfss 3055 | . . 3 | |
9 | 7, 8 | mpbi 144 | . 2 |
10 | cnvin 4916 | . 2 | |
11 | cnvin 4916 | . . . 4 | |
12 | 11 | cnveqi 4684 | . . 3 |
13 | inss2 3267 | . . . . 5 | |
14 | df-rel 4516 | . . . . 5 | |
15 | 13, 14 | mpbir 145 | . . . 4 |
16 | dfrel2 4959 | . . . 4 | |
17 | 15, 16 | mpbi 144 | . . 3 |
18 | 12, 17 | eqtr3i 2140 | . 2 |
19 | 9, 10, 18 | 3eqtr2i 2144 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1316 cvv 2660 cin 3040 wss 3041 cxp 4507 ccnv 4508 wrel 4514 |
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 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-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 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-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 |
This theorem is referenced by: cnvcnv2 4962 cnvcnvss 4963 structcnvcnv 11886 strslfv2d 11912 |
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