Theorem relco 5200

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 4702	. 2 |- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) }
2	1	relopabi 4821	1 |- Rel ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1516   class class class wbr 4059   o. ccom 4697   Rel wrel 4698

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-opab 4122  df-xp 4699  df-rel 4700  df-co 4702

This theorem is referenced by:  dfco2  5201  resco  5206  coiun  5211  cocnvcnv2  5213  cores2  5214  co02  5215  co01  5216  coi1  5217  coass  5220  cossxp  5224  funco  5330  fmptco  5769  cofunexg  6217  dftpos4  6372  znleval  14530