Definition df-co 5709

Description: Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 30470) because (cos‘0) = 1 (see cos0 16198) and (exp‘1) = e (see df-e 16116). 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	⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧

Detailed syntax breakdown of Definition df-co

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3	1, 2	ccom 5704	. 2 class (𝐴 ∘ 𝐵)
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1536	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1536	. . . . . 6 class 𝑧
8	5, 7, 2	wbr 5166	. . . . 5 wff 𝑥𝐵𝑧
9		vy	. . . . . . 7 setvar 𝑦
10	9	cv 1536	. . . . . 6 class 𝑦
11	7, 10, 1	wbr 5166	. . . . 5 wff 𝑧𝐴𝑦
12	8, 11	wa 395	. . . 4 wff (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)
13	12, 6	wex 1777	. . 3 wff ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)
14	13, 4, 9	copab 5228	. 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
15	3, 14	wceq 1537	1 wff (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}

This definition is referenced by:  coss1  5880  coss2  5881  nfco  5890  brcog  5891  cnvco  5910  relco  6138  coundi  6278  coundir  6279  cores  6280  xpco  6320  dffun2OLDOLD  6585  funco  6618  xpcomco  9128  coss12d  15021  xpcogend  15023  trclublem  15044  rtrclreclem3  15109  bj-opabco  37154  bj-xpcossxp  37155  dfcoss3  38370