Theorem relco 5045

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 4556	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
2	1	relopabi 4673	1 ⊢ Rel (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 103  ∃wex 1469   class class class wbr 3937   ∘ ccom 4551  Rel wrel 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-opab 3998  df-xp 4553  df-rel 4554  df-co 4556

This theorem is referenced by:  dfco2  5046  resco  5051  coiun  5056  cocnvcnv2  5058  cores2  5059  co02  5060  co01  5061  coi1  5062  coass  5065  cossxp  5069  funco  5171  fmptco  5594  cofunexg  6017  dftpos4  6168