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Theorem relco 5413
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 4922 . 2  |-  ( A  o.  B )  =  { <. x ,  y
>.  |  E. z
( x B z  /\  z A y ) }
21relopabi 5040 1  |-  Rel  ( A  o.  B )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367   E.wex 1620   class class class wbr 4367    o. ccom 4917   Rel wrel 4918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-opab 4426  df-xp 4919  df-rel 4920  df-co 4922
This theorem is referenced by:  dfco2  5414  resco  5419  coeq0  5424  coiun  5425  cocnvcnv2  5427  cores2  5428  co02  5429  co01  5430  coi1  5431  coass  5434  cossxp  5438  fmptco  5966  cofunexg  6663  dftpos4  6892  wunco  9022  relexprelg  12873  relexpaddg  12888  imasless  14947  znleval  18684  metustexhalfOLD  21151  metustexhalf  21152  fcoinver  27593  fmptcof2  27643  dfpo2  29350  cnvco1  29355  cnvco2  29356  opelco3  29373  txpss3v  29681  sscoid  29716  sblpnf  31358  coiun1  38189  relexpaddss  38223
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