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Definition df-co 5599
Description: Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 28811) because (cos‘0) = 1 (see cos0 15868) and (exp‘1) = e (see df-e 15787). 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 5594 . 2 class (𝐴𝐵)
4 vx . . . . . . 7 setvar 𝑥
54cv 1538 . . . . . 6 class 𝑥
6 vz . . . . . . 7 setvar 𝑧
76cv 1538 . . . . . 6 class 𝑧
85, 7, 2wbr 5075 . . . . 5 wff 𝑥𝐵𝑧
9 vy . . . . . . 7 setvar 𝑦
109cv 1538 . . . . . 6 class 𝑦
117, 10, 1wbr 5075 . . . . 5 wff 𝑧𝐴𝑦
128, 11wa 396 . . . 4 wff (𝑥𝐵𝑧𝑧𝐴𝑦)
1312, 6wex 1782 . . 3 wff 𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)
1413, 4, 9copab 5137 . 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
153, 14wceq 1539 1 wff (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
Colors of variables: wff setvar class
This definition is referenced by:  coss1  5767  coss2  5768  nfco  5777  brcog  5778  cnvco  5797  relco  6019  coundi  6155  coundir  6156  cores  6157  xpco  6196  dffun2OLD  6448  funco  6481  xpcomco  8858  coss12d  14692  xpcogend  14694  trclublem  14715  rtrclreclem3  14780  bj-opabco  35368  bj-xpcossxp  35369  dfcoss3  36547
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