Definition df-omul 3107

Description: Define the ordinal multiplication operation.

Assertion

Ref	Expression
df-omul	⊢ ·o = {⟨⟨x, y⟩, z⟩∣((x ∈ On ∧ y ∈ On) ∧ z = (rec({⟨w, v⟩∣v = (w +o x)}, ∅) ‘y))}

Distinct variable group(s):   x,y,z,w,v

Detailed syntax breakdown of Definition df-omul

Step	Hyp	Ref	Expression
1		comu 3102	. 2 class ·o
2		vx	. . . . . . 7 set x
3	2	cv 1089	. . . . . 6 class x
4		con0 2199	. . . . . 6 class On
5	3, 4	wcel 1092	. . . . 5 wff x ∈ On
6		vy	. . . . . . 7 set y
7	6	cv 1089	. . . . . 6 class y
8	7, 4	wcel 1092	. . . . 5 wff y ∈ On
9	5, 8	wa 196	. . . 4 wff (x ∈ On ∧ y ∈ On)
10		vz	. . . . . 6 set z
11	10	cv 1089	. . . . 5 class z
12		vv	. . . . . . . . . 10 set v
13	12	cv 1089	. . . . . . . . 9 class v
14		vw	. . . . . . . . . . 11 set w
15	14	cv 1089	. . . . . . . . . 10 class w
16		coa 3101	. . . . . . . . . 10 class +o
17	15, 3, 16	co 3001	. . . . . . . . 9 class (w +o x)
18	13, 17	wceq 1091	. . . . . . . 8 wff v = (w +o x)
19	18, 14, 12	copab 2055	. . . . . . 7 class {⟨w, v⟩∣v = (w +o x)}
20		c0 1707	. . . . . . 7 class ∅
21	19, 20	crdg 2969	. . . . . 6 class rec({⟨w, v⟩∣v = (w +o x)}, ∅)
22	7, 21	cfv 2422	. . . . 5 class (rec({⟨w, v⟩∣v = (w +o x)}, ∅) ‘y)
23	11, 22	wceq 1091	. . . 4 wff z = (rec({⟨w, v⟩∣v = (w +o x)}, ∅) ‘y)
24	9, 23	wa 196	. . 3 wff ((x ∈ On ∧ y ∈ On) ∧ z = (rec({⟨w, v⟩∣v = (w +o x)}, ∅) ‘y))
25	24, 2, 6, 10	copab2 3002	. 2 class {⟨⟨x, y⟩, z⟩∣((x ∈ On ∧ y ∈ On) ∧ z = (rec({⟨w, v⟩∣v = (w +o x)}, ∅) ‘y))}
26	1, 25	wceq 1091	1 wff ·o = {⟨⟨x, y⟩, z⟩∣((x ∈ On ∧ y ∈ On) ∧ z = (rec({⟨w, v⟩∣v = (w +o x)}, ∅) ‘y))}

This definition is referenced by:  fnom 3118  omv 3120

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