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Theorem relco 5359
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 4878 . 2  |-  ( A  o.  B )  =  { <. x ,  y
>.  |  E. z
( x B z  /\  z A y ) }
21relopabi 4991 1  |-  Rel  ( A  o.  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550   class class class wbr 4204    o. ccom 4873   Rel wrel 4874
This theorem is referenced by:  dfco2  5360  resco  5365  coiun  5370  cocnvcnv2  5372  cores2  5373  co02  5374  co01  5375  coi1  5376  coass  5379  cossxp  5383  fmptco  5892  cofunexg  5950  dftpos4  6489  wunco  8597  imasless  13753  znleval  16823  metustexhalfOLD  18581  metustexhalf  18582  fmptcof2  24064  dfpo2  25367  cnvco1  25372  cnvco2  25373  txpss3v  25673  sscoid  25708  coeq0  26747  sblpnf  27454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4875  df-rel 4876  df-co 4878
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