Theorem relco 5227

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 4728	. 2 |- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) }
2	1	relopabi 4847	1 |- Rel ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1538   class class class wbr 4083   o. ccom 4723   Rel wrel 4724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146  df-xp 4725  df-rel 4726  df-co 4728

This theorem is referenced by:  dfco2  5228  resco  5233  coiun  5238  cocnvcnv2  5240  cores2  5241  co02  5242  co01  5243  coi1  5244  coass  5247  cossxp  5251  funco  5358  fmptco  5801  cofunexg  6254  dftpos4  6409  znleval  14617