Theorem relco 5231

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 4730	. 2 |- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) }
2	1	relopabi 4851	1 |- Rel ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1538   class class class wbr 4084   o. ccom 4725   Rel wrel 4726

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 4203  ax-pow 4260  ax-pr 4295

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 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-opab 4147  df-xp 4727  df-rel 4728  df-co 4730

This theorem is referenced by:  dfco2  5232  resco  5237  coiun  5242  cocnvcnv2  5244  cores2  5245  co02  5246  co01  5247  coi1  5248  coass  5251  cossxp  5255  funco  5362  fmptco  5807  cofunexg  6264  dftpos4  6422  znleval  14654