Theorem relco 5550

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 5047	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
2	1	relopabi 5167	1 ⊢ Rel (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 383  ∃wex 1695   class class class wbr 4583   ∘ ccom 5042  Rel wrel 5043

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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

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-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 4644  df-xp 5044  df-rel 5045  df-co 5047

This theorem is referenced by:  dfco2  5551  resco  5556  coeq0  5561  coiun  5562  cocnvcnv2  5564  cores2  5565  co02  5566  co01  5567  coi1  5568  coass  5571  cossxp  5575  fmptco  6303  cofunexg  7023  dftpos4  7258  wunco  9434  relexprelg  13626  relexpaddg  13641  imasless  16023  znleval  19722  metustexhalf  22171  fcoinver  28798  fmptcof2  28839  dfpo2  30898  cnvco1  30903  cnvco2  30904  opelco3  30923  txpss3v  31155  sscoid  31190  cononrel1  36919  cononrel2  36920  coiun1  36963  relexpaddss  37029  brco2f1o  37350  brco3f1o  37351  neicvgnvor  37434  sblpnf  37531