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Definition df-wdom 9602
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 8985) implies weak dominance (over ZF). The principle asserting the converse is known as the partition principle and is independent of ZF. Theorem fodom 10560 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 9601 . 2 class *
2 vx . . . . . 6 setvar 𝑥
32cv 1535 . . . . 5 class 𝑥
4 c0 4338 . . . . 5 class
53, 4wceq 1536 . . . 4 wff 𝑥 = ∅
6 vy . . . . . . 7 setvar 𝑦
76cv 1535 . . . . . 6 class 𝑦
8 vz . . . . . . 7 setvar 𝑧
98cv 1535 . . . . . 6 class 𝑧
107, 3, 9wfo 6560 . . . . 5 wff 𝑧:𝑦onto𝑥
1110, 8wex 1775 . . . 4 wff 𝑧 𝑧:𝑦onto𝑥
125, 11wo 847 . . 3 wff (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)
1312, 2, 6copab 5209 . 2 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
141, 13wceq 1536 1 wff * = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
Colors of variables: wff setvar class
This definition is referenced by:  relwdom  9603  brwdom  9604
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