Theorem relco 5077

Description: A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)

Assertion

Ref	Expression
relco	|- Rel ( A o. 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 4588	. 2 |- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) }
2	1	relopabi 4705	1 |- Rel ( A o. B )

Syntax hints:   /\ wa 103   E.wex 1469   class class class wbr 3961   o. ccom 4583   Rel wrel 4584

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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-opab 4022  df-xp 4585  df-rel 4586  df-co 4588

This theorem is referenced by:  dfco2  5078  resco  5083  coiun  5088  cocnvcnv2  5090  cores2  5091  co02  5092  co01  5093  coi1  5094  coass  5097  cossxp  5101  funco  5203  fmptco  5626  cofunexg  6049  dftpos4  6200