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 5086   ∘ 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 3432  df-ss 3907  df-opab 5149  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  7077  cofunexg  7896  dftpos4  8189  ttrcltr  9631  ttrclco  9633  wunco  10650  relexprelg  14994  relexpaddg  15009  imasless  17498  znleval  21547  metustexhalf  24534  fcoinver  32692  fmptcof2  32748  cnvco1  35960  cnvco2  35961  opelco3  35976  txpss3v  36077  sscoid  36112  xrnss3v  38719  cononrel1  44042  cononrel2  44043  coiun1  44100  relexpaddss  44166  brco2f1o  44480  brco3f1o  44481  neicvgnvor  44564  sblpnf  44758  coxp  49323  xpco2  49347