Theorem relco 5852

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 5321	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
2	1	relopabi 5449	1 ⊢ Rel (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 385  ∃wex 1875   class class class wbr 4843   ∘ ccom 5316  Rel wrel 5317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-opab 4906  df-xp 5318  df-rel 5319  df-co 5321

This theorem is referenced by:  dfco2  5853  resco  5858  coeq0  5863  coiun  5864  cocnvcnv2  5866  cores2  5867  co02  5868  co01  5869  coi1  5870  coass  5873  cossxp  5877  fmptco  6623  cofunexg  7365  dftpos4  7609  wunco  9843  relexprelg  14119  relexpaddg  14134  imasless  16515  znleval  20224  metustexhalf  22689  fcoinver  29935  fmptcof2  29976  dfpo2  32159  cnvco1  32163  cnvco2  32164  opelco3  32190  txpss3v  32498  sscoid  32533  xrnss3v  34628  cononrel1  38683  cononrel2  38684  coiun1  38727  relexpaddss  38793  brco2f1o  39112  brco3f1o  39113  neicvgnvor  39196  sblpnf  39291