Definition df-wdom 9095

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 8550) implies weak dominance (over ZF). The principle asserting the converse is known as the partition principle and is independent of ZF. Theorem fodom 10016 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 9094	. 2 class ≼*
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1541	. . . . 5 class 𝑥
4		c0 4209	. . . . 5 class ∅
5	3, 4	wceq 1542	. . . 4 wff 𝑥 = ∅
6		vy	. . . . . . 7 setvar 𝑦
7	6	cv 1541	. . . . . 6 class 𝑦
8		vz	. . . . . . 7 setvar 𝑧
9	8	cv 1541	. . . . . 6 class 𝑧
10	7, 3, 9	wfo 6331	. . . . 5 wff 𝑧:𝑦–onto→𝑥
11	10, 8	wex 1786	. . . 4 wff ∃𝑧 𝑧:𝑦–onto→𝑥
12	5, 11	wo 846	. . 3 wff (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)
13	12, 2, 6	copab 5089	. 2 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
14	1, 13	wceq 1542	1 wff ≼* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}

This definition is referenced by:  relwdom  9096  brwdom  9097