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Theorem relco 5005
Description: A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
Assertion
Ref Expression
relco  |-  Rel  ( A  o.  B )

Proof of Theorem relco
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-co 4516 . 2  |-  ( A  o.  B )  =  { <. x ,  y
>.  |  E. z
( x B z  /\  z A y ) }
21relopabi 4633 1  |-  Rel  ( A  o.  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103   E.wex 1451   class class class wbr 3897    o. ccom 4511   Rel wrel 4512
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-xp 4513  df-rel 4514  df-co 4516
This theorem is referenced by:  dfco2  5006  resco  5011  coiun  5016  cocnvcnv2  5018  cores2  5019  co02  5020  co01  5021  coi1  5022  coass  5025  cossxp  5029  funco  5131  fmptco  5552  cofunexg  5975  dftpos4  6126
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