Definition df-co 2427

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 / instead of ∘, and calls the operation "relative product."

Assertion

Ref	Expression
df-co	⊢ (A ∘ B) = {⟨x, y⟩∣∃z(xBz ∧ zAy)}

Distinct variable group(s):   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 2414	. 2 class (A ∘ B)
4		vx	. . . . . . 7 set x
5	4	cv 1089	. . . . . 6 class x
6		vz	. . . . . . 7 set z
7	6	cv 1089	. . . . . 6 class z
8	5, 7, 2	wbr 2054	. . . . 5 wff xBz
9		vy	. . . . . . 7 set y
10	9	cv 1089	. . . . . 6 class y
11	7, 10, 1	wbr 2054	. . . . 5 wff zAy
12	8, 11	wa 196	. . . 4 wff (xBz ∧ zAy)
13	12, 6	wex 678	. . 3 wff ∃z(xBz ∧ zAy)
14	13, 4, 9	copab 2055	. 2 class {⟨x, y⟩∣∃z(xz ∧ zAy)}
15	3, 14	wceq 1091	1 wff (A ∘ B) = {⟨x, y⟩∣∃z(xBz ∧ zAy)}

This definition is referenced by:  coeq1 2502  coeq2 2503  hbco 2508  opelco 2509  cnvco 2520  cotr 2625  relco 2658  dffun2 2674

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