Theorem relco 2658

Description: A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25.

Assertion

Ref	Expression
relco	|- Rel (A o. B)

Proof of Theorem relco

Step	Hyp	Ref	Expression
1		relopab 2494	. 2 |- Rel {<.x, y>. | E.z(xBz /\ zAy)}
2		df-co 2427	. . 3 |- (A o. B) = {<.x, y>. | E.z(xBz /\ zAy)}
3		releq 2477	. . 3 |- ((A o. B) = {<.x, y>. | E.z(xBz /\ zAy)} -> (Rel (A o. B) <-> Rel {<.x, y>. | E.z(xBz /\ zAy)}))
4	2, 3	ax-mp 6	. 2 |- (Rel (A o. B) <-> Rel {<.x, y>. | E.z(xBz /\ zAy)})
5	1, 4	mpbir 165	1 |- Rel (A o. B)

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   class class class wbr 2054  {copab 2055   o. ccom 2414  Rel wrel 2415

This theorem is referenced by:  cores 2659  co02 2663  co01 2664  coi1 2665  coass 2667  coexg 2671  funco 2696  fcoi1 2765  fcoi2 2766

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425  df-co 2427

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