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Theorem cnvco 4548
 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.)
Assertion
Ref Expression
cnvco

Proof of Theorem cnvco
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exancom 1515 . . . 4
2 vex 2577 . . . . 5
3 vex 2577 . . . . 5
42, 3brco 4534 . . . 4
5 vex 2577 . . . . . . 7
63, 5brcnv 4546 . . . . . 6
75, 2brcnv 4546 . . . . . 6
86, 7anbi12i 441 . . . . 5
98exbii 1512 . . . 4
101, 4, 93bitr4i 205 . . 3
1110opabbii 3852 . 2
12 df-cnv 4381 . 2
13 df-co 4382 . 2
1411, 12, 133eqtr4i 2086 1
 Colors of variables: wff set class Syntax hints:   wa 101   wceq 1259  wex 1397   class class class wbr 3792  copab 3845  ccnv 4372   ccom 4377 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-cnv 4381  df-co 4382 This theorem is referenced by:  rncoss  4630  rncoeq  4633  dmco  4857  cores2  4861  co01  4863  coi2  4865  relcnvtr  4868  dfdm2  4880  f1co  5129  cofunex2g  5767
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