Theorem relco 6068

Description: A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)

Assertion

Ref	Expression
relco	⊢ Rel (𝐴 ∘ 𝐵)

Proof of Theorem relco

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-co 5634	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
2	1	relopabiv 5770	1 ⊢ Rel (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 395  ∃wex 1781   class class class wbr 5099   ∘ ccom 5629  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-ss 3919  df-opab 5162  df-xp 5631  df-rel 5632  df-co 5634

This theorem is referenced by:  cotrg  6069  dfco2  6204  resco  6209  coeq0  6215  coiun  6216  cocnvcnv2  6218  cores2  6219  co02  6220  co01  6221  coi1  6222  coass  6225  cossxp  6231  dfpo2  6255  fmptco  7076  cofunexg  7895  dftpos4  8189  ttrcltr  9629  ttrclco  9631  wunco  10648  relexprelg  14965  relexpaddg  14980  imasless  17465  znleval  21513  metustexhalf  24504  fcoinver  32661  fmptcof2  32717  cnvco1  35934  cnvco2  35935  opelco3  35950  txpss3v  36051  sscoid  36086  xrnss3v  38553  cononrel1  43871  cononrel2  43872  coiun1  43929  relexpaddss  43995  brco2f1o  44309  brco3f1o  44310  neicvgnvor  44393  sblpnf  44587  coxp  49114  xpco2  49138