ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-co GIF version

Definition df-co 4635
Description: Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. Note that Definition 7 of [Suppes] p. 63 reverses 𝐴 and 𝐵, uses a slash instead of , and calls the operation "relative product". (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
df-co (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧

Detailed syntax breakdown of Definition df-co
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
31, 2ccom 4630 . 2 class (𝐴𝐵)
4 vx . . . . . . 7 setvar 𝑥
54cv 1352 . . . . . 6 class 𝑥
6 vz . . . . . . 7 setvar 𝑧
76cv 1352 . . . . . 6 class 𝑧
85, 7, 2wbr 4003 . . . . 5 wff 𝑥𝐵𝑧
9 vy . . . . . . 7 setvar 𝑦
109cv 1352 . . . . . 6 class 𝑦
117, 10, 1wbr 4003 . . . . 5 wff 𝑧𝐴𝑦
128, 11wa 104 . . . 4 wff (𝑥𝐵𝑧𝑧𝐴𝑦)
1312, 6wex 1492 . . 3 wff 𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)
1413, 4, 9copab 4063 . 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
153, 14wceq 1353 1 wff (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
Colors of variables: wff set class
This definition is referenced by:  coss1  4782  coss2  4783  nfco  4792  elco  4793  brcog  4794  cnvco  4812  cotr  5010  relco  5127  coundi  5130  coundir  5131  cores  5132  xpcom  5175  dffun2  5226  funco  5256  xpcomco  6825
  Copyright terms: Public domain W3C validator