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Definition df-wdom 9580
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 8957) implies weak dominance (over ZF). The principle asserting the converse is known as the partition principle and is independent of ZF. Theorem fodom 10538 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:𝑦onto𝑥)}
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-wdom
StepHypRef Expression
1 cwdom 9579 . 2 class *
2 vx . . . . . 6 setvar 𝑥
32cv 1533 . . . . 5 class 𝑥
4 c0 4318 . . . . 5 class
53, 4wceq 1534 . . . 4 wff 𝑥 = ∅
6 vy . . . . . . 7 setvar 𝑦
76cv 1533 . . . . . 6 class 𝑦
8 vz . . . . . . 7 setvar 𝑧
98cv 1533 . . . . . 6 class 𝑧
107, 3, 9wfo 6540 . . . . 5 wff 𝑧:𝑦onto𝑥
1110, 8wex 1774 . . . 4 wff 𝑧 𝑧:𝑦onto𝑥
125, 11wo 846 . . 3 wff (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)
1312, 2, 6copab 5204 . 2 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
141, 13wceq 1534 1 wff * = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
Colors of variables: wff setvar class
This definition is referenced by:  relwdom  9581  brwdom  9582
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