Theorem relco 6137

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 5589	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
2	1	relopabiv 5719	1 ⊢ Rel (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 395  ∃wex 1783   class class class wbr 5070   ∘ ccom 5584  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-opab 5133  df-xp 5586  df-rel 5587  df-co 5589

This theorem is referenced by:  dfco2  6138  resco  6143  coeq0  6148  coiun  6149  cocnvcnv2  6151  cores2  6152  co02  6153  co01  6154  coi1  6155  coass  6158  cossxp  6164  dfpo2  6188  fmptco  6983  cofunexg  7765  dftpos4  8032  wunco  10420  relexprelg  14677  relexpaddg  14692  imasless  17168  znleval  20674  metustexhalf  23618  fcoinver  30847  fmptcof2  30896  cnvco1  33632  cnvco2  33633  opelco3  33655  ttrcltr  33702  ttrclco  33704  txpss3v  34107  sscoid  34142  xrnss3v  36429  cononrel1  41091  cononrel2  41092  coiun1  41149  relexpaddss  41215  brco2f1o  41531  brco3f1o  41532  neicvgnvor  41615  sblpnf  41817