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Definition df-wdom 8505
Description: A set is weakly dominated by a "larger" set iff the "larger" set can be mapped onto the "smaller" set or the smaller set is empty; equivalently if the smaller set can be placed into bijection with some partition of the larger set. When choice is assumed (as fodom 9382), this coincides with the 1-1 definition df-dom 7999; however, it is not known whether this is a choice-equivalent or a strictly weaker form. Some discussion of this question can be found at http://boolesrings.org/asafk/2014/on-the-partition-principle/. (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 8503 . 2 class *
2 vx . . . . . 6 setvar 𝑥
32cv 1522 . . . . 5 class 𝑥
4 c0 3948 . . . . 5 class
53, 4wceq 1523 . . . 4 wff 𝑥 = ∅
6 vy . . . . . . 7 setvar 𝑦
76cv 1522 . . . . . 6 class 𝑦
8 vz . . . . . . 7 setvar 𝑧
98cv 1522 . . . . . 6 class 𝑧
107, 3, 9wfo 5924 . . . . 5 wff 𝑧:𝑦onto𝑥
1110, 8wex 1744 . . . 4 wff 𝑧 𝑧:𝑦onto𝑥
125, 11wo 382 . . 3 wff (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)
1312, 2, 6copab 4745 . 2 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
141, 13wceq 1523 1 wff * = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
Colors of variables: wff setvar class
This definition is referenced by:  relwdom  8512  brwdom  8513
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