Definition df-co 5659

Description: Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 31039) because (cos‘0) = 1 (see cos0 16318) and (exp‘1) = e (see df-e 16234). 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 5654	. 2 class (𝐴 ∘ 𝐵)
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1568	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1568	. . . . . 6 class 𝑧
8	5, 7, 2	wbr 5102	. . . . 5 wff 𝑥𝐵𝑧
9		vy	. . . . . . 7 setvar 𝑦
10	9	cv 1568	. . . . . 6 class 𝑦
11	7, 10, 1	wbr 5102	. . . . 5 wff 𝑧𝐴𝑦
12	8, 11	wa 401	. . . 4 wff (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)
13	12, 6	wex 1811	. . 3 wff ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)
14	13, 4, 9	copab 5166	. 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
15	3, 14	wceq 1569	1 wff (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}

This definition is used by:  coss1  5832  coss2  5833  nfco  5842  brcog  5843  cnvco  5866  relco  6103  coundi  6248  coundir  6249  cores  6250  xpco  6292  funco  6580  xpcomco  9086  coss12d  15125  xpcogend  15127  trclublem  15148  rtrclreclem3  15213  dfsuccf2  36747  elco  38002  bj-opabco  38152  bj-xpcossxp  38153  dfcoss3  39479