Definition df-co 5628

Description: Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 30382) because (cos‘0) = 1 (see cos0 16059) and (exp‘1) = e (see df-e 15975). 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 5623	. 2 class (𝐴 ∘ 𝐵)
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1539	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1539	. . . . . 6 class 𝑧
8	5, 7, 2	wbr 5092	. . . . 5 wff 𝑥𝐵𝑧
9		vy	. . . . . . 7 setvar 𝑦
10	9	cv 1539	. . . . . 6 class 𝑦
11	7, 10, 1	wbr 5092	. . . . 5 wff 𝑧𝐴𝑦
12	8, 11	wa 395	. . . 4 wff (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)
13	12, 6	wex 1779	. . 3 wff ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)
14	13, 4, 9	copab 5154	. 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
15	3, 14	wceq 1540	1 wff (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}

This definition is referenced by:  coss1  5798  coss2  5799  nfco  5808  brcog  5809  cnvco  5828  relco  6059  coundi  6196  coundir  6197  cores  6198  xpco  6237  funco  6522  xpcomco  8984  coss12d  14879  xpcogend  14881  trclublem  14902  rtrclreclem3  14967  bj-opabco  37172  bj-xpcossxp  37173  dfcoss3  38401