Theorem relco 3490

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

Assertion

Ref	Expression
relco	⊢ Rel (A ∘ B)

Proof of Theorem relco

Step	Hyp	Ref	Expression
1		relopab 3272	. 2 ⊢ Rel {⟨x, y⟩∣∃z(xBz ⋀ zAy)}
2		df-co 3193	. . 3 ⊢ (A ∘ B) = {⟨x, y⟩∣∃z(xBz ⋀ zAy)}
3	2	releqi 3250	. 2 ⊢ (Rel (A ∘ B) ↔ Rel {⟨x, y⟩∣∃z(xBz ⋀ zAy)})
4	1, 3	mpbir 190	1 ⊢ Rel (A ∘ B)

Syntax hints:   ⋀ wa 223  ∃wex 982   class class class wbr 2624  {copab 2671   ∘ ccom 3180  Rel wrel 3181

This theorem is referenced by:  cores 3505  resco 3506  cocnvcnv2 3512  cores2 3513  co02 3514  co01 3515  coi1 3516  coass 3518  coexg 3530  funco 3556  cofunexg 3586  fcoi1 3651  fcoi2 3652  cncfmet1 7903  abscncfALT 8340

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-xp 3190  df-rel 3191  df-co 3193

Copyright terms: Public domain