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Theorem relco 5536
Description: A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
Assertion
Ref Expression
relco Rel (𝐴𝐵)

Proof of Theorem relco
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-co 5037 . 2 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
21relopabi 5156 1 Rel (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 383  wex 1695   class class class wbr 4578  ccom 5032  Rel wrel 5033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4639  df-xp 5034  df-rel 5035  df-co 5037
This theorem is referenced by:  dfco2  5537  resco  5542  coeq0  5547  coiun  5548  cocnvcnv2  5550  cores2  5551  co02  5552  co01  5553  coi1  5554  coass  5557  cossxp  5561  fmptco  6288  cofunexg  7001  dftpos4  7236  wunco  9412  relexprelg  13575  relexpaddg  13590  imasless  15972  znleval  19670  metustexhalf  22119  fcoinver  28592  fmptcof2  28633  dfpo2  30692  cnvco1  30697  cnvco2  30698  opelco3  30717  txpss3v  30949  sscoid  30984  cononrel1  36713  cononrel2  36714  coiun1  36757  relexpaddss  36823  brco2f1o  37144  brco3f1o  37145  neicvgnvor  37228  sblpnf  37325
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