Definition df-co 4672

Description: Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. Note that Definition 7 of [Suppes] p. 63 reverses 𝐴 and 𝐵, uses a slash 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 4667	. 2 class (𝐴 ∘ 𝐵)
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1363	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1363	. . . . . 6 class 𝑧
8	5, 7, 2	wbr 4033	. . . . 5 wff 𝑥𝐵𝑧
9		vy	. . . . . . 7 setvar 𝑦
10	9	cv 1363	. . . . . 6 class 𝑦
11	7, 10, 1	wbr 4033	. . . . 5 wff 𝑧𝐴𝑦
12	8, 11	wa 104	. . . 4 wff (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)
13	12, 6	wex 1506	. . 3 wff ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)
14	13, 4, 9	copab 4093	. 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
15	3, 14	wceq 1364	1 wff (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}

This definition is referenced by:  coss1  4821  coss2  4822  nfco  4831  elco  4832  brcog  4833  cnvco  4851  cotr  5051  relco  5168  coundi  5171  coundir  5172  cores  5173  xpcom  5216  dffun2  5268  funco  5298  xpcomco  6885