Definition df-wdom 9518

Description: A set is weakly dominated by a "larger" set if the "larger" set can be mapped onto the "smaller" set or the smaller set is empty, or equivalently, if the smaller set can be placed into bijection with some partition of the larger set. Dominance (df-dom 8920) implies weak dominance (over ZF). The principle asserting the converse is known as the partition principle and is independent of ZF. Theorem fodom 10476 proves that the axiom of choice implies the partition principle (over ZF). It is not known whether the partition principle is equivalent to the axiom of choice (over ZF), although it is know to imply dependent choice. (Contributed by Stefan O'Rear, 11-Feb-2015.)

Assertion

Ref	Expression
df-wdom	⊢ ≼* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-wdom

Step	Hyp	Ref	Expression
1		cwdom 9517	. 2 class ≼*
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1539	. . . . 5 class 𝑥
4		c0 4296	. . . . 5 class ∅
5	3, 4	wceq 1540	. . . 4 wff 𝑥 = ∅
6		vy	. . . . . . 7 setvar 𝑦
7	6	cv 1539	. . . . . 6 class 𝑦
8		vz	. . . . . . 7 setvar 𝑧
9	8	cv 1539	. . . . . 6 class 𝑧
10	7, 3, 9	wfo 6509	. . . . 5 wff 𝑧:𝑦–onto→𝑥
11	10, 8	wex 1779	. . . 4 wff ∃𝑧 𝑧:𝑦–onto→𝑥
12	5, 11	wo 847	. . 3 wff (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)
13	12, 2, 6	copab 5169	. 2 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
14	1, 13	wceq 1540	1 wff ≼* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}

This definition is referenced by:  relwdom  9519  brwdom  9520