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Definition df-co 5558
Description: Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 28145) because (cos‘0) = 1 (see cos0 15493) and (exp‘1) = e (see df-e 15412). 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 5553 . 2 class (𝐴𝐵)
4 vx . . . . . . 7 setvar 𝑥
54cv 1527 . . . . . 6 class 𝑥
6 vz . . . . . . 7 setvar 𝑧
76cv 1527 . . . . . 6 class 𝑧
85, 7, 2wbr 5058 . . . . 5 wff 𝑥𝐵𝑧
9 vy . . . . . . 7 setvar 𝑦
109cv 1527 . . . . . 6 class 𝑦
117, 10, 1wbr 5058 . . . . 5 wff 𝑧𝐴𝑦
128, 11wa 396 . . . 4 wff (𝑥𝐵𝑧𝑧𝐴𝑦)
1312, 6wex 1771 . . 3 wff 𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)
1413, 4, 9copab 5120 . 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
153, 14wceq 1528 1 wff (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
Colors of variables: wff setvar class
This definition is referenced by:  coss1  5720  coss2  5721  nfco  5730  brcog  5731  cnvco  5750  cotrg  5965  relco  6091  coundi  6094  coundir  6095  cores  6096  xpco  6134  dffun2  6359  funco  6389  xpcomco  8596  coss12d  14322  xpcogend  14324  trclublem  14345  rtrclreclem3  14409  dfcoss3  35544
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