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Definition df-co 4555
 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 4550 . 2 class (𝐴𝐵)
4 vx . . . . . . 7 setvar 𝑥
54cv 1331 . . . . . 6 class 𝑥
6 vz . . . . . . 7 setvar 𝑧
76cv 1331 . . . . . 6 class 𝑧
85, 7, 2wbr 3936 . . . . 5 wff 𝑥𝐵𝑧
9 vy . . . . . . 7 setvar 𝑦
109cv 1331 . . . . . 6 class 𝑦
117, 10, 1wbr 3936 . . . . 5 wff 𝑧𝐴𝑦
128, 11wa 103 . . . 4 wff (𝑥𝐵𝑧𝑧𝐴𝑦)
1312, 6wex 1469 . . 3 wff 𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)
1413, 4, 9copab 3995 . 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
153, 14wceq 1332 1 wff (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
 Colors of variables: wff set class This definition is referenced by:  coss1  4701  coss2  4702  nfco  4711  elco  4712  brcog  4713  cnvco  4731  cotr  4927  relco  5044  coundi  5047  coundir  5048  cores  5049  xpcom  5092  dffun2  5140  funco  5170  xpcomco  6727
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