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Mirrors > Home > ILE Home > Th. List > cnvco | Unicode version |
Description: Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exancom 1588 | . . . 4 | |
2 | vex 2715 | . . . . 5 | |
3 | vex 2715 | . . . . 5 | |
4 | 2, 3 | brco 4759 | . . . 4 |
5 | vex 2715 | . . . . . . 7 | |
6 | 3, 5 | brcnv 4771 | . . . . . 6 |
7 | 5, 2 | brcnv 4771 | . . . . . 6 |
8 | 6, 7 | anbi12i 456 | . . . . 5 |
9 | 8 | exbii 1585 | . . . 4 |
10 | 1, 4, 9 | 3bitr4i 211 | . . 3 |
11 | 10 | opabbii 4033 | . 2 |
12 | df-cnv 4596 | . 2 | |
13 | df-co 4597 | . 2 | |
14 | 11, 12, 13 | 3eqtr4i 2188 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1335 wex 1472 class class class wbr 3967 copab 4026 ccnv 4587 ccom 4592 |
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-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-cnv 4596 df-co 4597 |
This theorem is referenced by: rncoss 4858 rncoeq 4861 dmco 5096 cores2 5100 co01 5102 coi2 5104 relcnvtr 5107 dfdm2 5122 f1co 5389 cofunex2g 6062 caseinj 7035 djuinj 7052 cnco 12691 hmeoco 12786 |
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