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Theorem relco 5340
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 4848 . 2  |-  ( A  o.  B )  =  { <. x ,  y
>.  |  E. z
( x B z  /\  z A y ) }
21relopabi 4964 1  |-  Rel  ( A  o.  B )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376   E.wex 1671   class class class wbr 4395    o. ccom 4843   Rel wrel 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845  df-rel 4846  df-co 4848
This theorem is referenced by:  dfco2  5341  resco  5346  coeq0  5351  coiun  5352  cocnvcnv2  5354  cores2  5355  co02  5356  co01  5357  coi1  5358  coass  5361  cossxp  5365  fmptco  6072  cofunexg  6776  dftpos4  7010  wunco  9176  relexprelg  13178  relexpaddg  13193  imasless  15524  znleval  19202  metustexhalf  21649  fcoinver  28290  fmptcof2  28334  dfpo2  30466  cnvco1  30471  cnvco2  30472  opelco3  30491  txpss3v  30716  sscoid  30751  cononrel1  36271  cononrel2  36272  coiun1  36315  relexpaddss  36381  sblpnf  36728
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