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Definition df-co 5267
Description: Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 27598) because (cos‘0) = 1 (see cos0 15071) and (exp‘1) = e (see df-e 14990). Note that Definition 7 of [Suppes] p. 63 reverses 𝐴 and 𝐵, uses / 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 5262 . 2 class (𝐴𝐵)
4 vx . . . . . . 7 setvar 𝑥
54cv 1623 . . . . . 6 class 𝑥
6 vz . . . . . . 7 setvar 𝑧
76cv 1623 . . . . . 6 class 𝑧
85, 7, 2wbr 4796 . . . . 5 wff 𝑥𝐵𝑧
9 vy . . . . . . 7 setvar 𝑦
109cv 1623 . . . . . 6 class 𝑦
117, 10, 1wbr 4796 . . . . 5 wff 𝑧𝐴𝑦
128, 11wa 383 . . . 4 wff (𝑥𝐵𝑧𝑧𝐴𝑦)
1312, 6wex 1845 . . 3 wff 𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)
1413, 4, 9copab 4856 . 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
153, 14wceq 1624 1 wff (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
Colors of variables: wff setvar class
This definition is referenced by:  coss1  5425  coss2  5426  nfco  5435  brcog  5436  cnvco  5455  cotrg  5657  relco  5786  coundi  5789  coundir  5790  cores  5791  xpco  5828  dffun2  6051  funco  6081  xpcomco  8207  coss12d  13904  xpcogend  13906  trclublem  13927  rtrclreclem3  13991  dfcoss3  34487
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