Theorem relco 5281

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 4778	. 2 |- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) }
2	1	relopabi 4900	1 |- Rel ( A o. B )

Syntax hints:   /\ wa 104   E.wex 1545   class class class wbr 4125   o. ccom 4773   Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188  df-xp 4775  df-rel 4776  df-co 4778

This theorem is referenced by:  dfco2  5282  resco  5287  coiun  5292  cocnvcnv2  5294  cores2  5295  co02  5296  co01  5297  coi1  5298  coass  5301  cossxp  5305  funco  5412  fmptco  5865  cofunexg  6328  dftpos4  6524  znleval  14960