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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 4966 | . . . . 5 | |
2 | df-rel 4595 | . . . . 5 | |
3 | 1, 2 | mpbi 144 | . . . 4 |
4 | relxp 4697 | . . . . 5 | |
5 | dfrel2 5038 | . . . . 5 | |
6 | 4, 5 | mpbi 144 | . . . 4 |
7 | 3, 6 | sseqtrri 3163 | . . 3 |
8 | dfss 3116 | . . 3 | |
9 | 7, 8 | mpbi 144 | . 2 |
10 | cnvin 4995 | . 2 | |
11 | cnvin 4995 | . . . 4 | |
12 | 11 | cnveqi 4763 | . . 3 |
13 | inss2 3329 | . . . . 5 | |
14 | df-rel 4595 | . . . . 5 | |
15 | 13, 14 | mpbir 145 | . . . 4 |
16 | dfrel2 5038 | . . . 4 | |
17 | 15, 16 | mpbi 144 | . . 3 |
18 | 12, 17 | eqtr3i 2180 | . 2 |
19 | 9, 10, 18 | 3eqtr2i 2184 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1335 cvv 2712 cin 3101 wss 3102 cxp 4586 ccnv 4587 wrel 4593 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4028 df-xp 4594 df-rel 4595 df-cnv 4596 |
This theorem is referenced by: cnvcnv2 5041 cnvcnvss 5042 structcnvcnv 12276 strslfv2d 12302 |
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