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Mirrors > Home > MPE Home > Th. List > df-wdom | Structured version Visualization version GIF version |
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 9666), this coincides with the 1-1 definition df-dom 8230; 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.) |
Ref | Expression |
---|---|
df-wdom | ⊢ ≼* = {〈𝑥, 𝑦〉 ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cwdom 8738 | . 2 class ≼* | |
2 | vx | . . . . . 6 setvar 𝑥 | |
3 | 2 | cv 1655 | . . . . 5 class 𝑥 |
4 | c0 4146 | . . . . 5 class ∅ | |
5 | 3, 4 | wceq 1656 | . . . 4 wff 𝑥 = ∅ |
6 | vy | . . . . . . 7 setvar 𝑦 | |
7 | 6 | cv 1655 | . . . . . 6 class 𝑦 |
8 | vz | . . . . . . 7 setvar 𝑧 | |
9 | 8 | cv 1655 | . . . . . 6 class 𝑧 |
10 | 7, 3, 9 | wfo 6125 | . . . . 5 wff 𝑧:𝑦–onto→𝑥 |
11 | 10, 8 | wex 1878 | . . . 4 wff ∃𝑧 𝑧:𝑦–onto→𝑥 |
12 | 5, 11 | wo 878 | . . 3 wff (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥) |
13 | 12, 2, 6 | copab 4937 | . 2 class {〈𝑥, 𝑦〉 ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)} |
14 | 1, 13 | wceq 1656 | 1 wff ≼* = {〈𝑥, 𝑦〉 ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)} |
Colors of variables: wff setvar class |
This definition is referenced by: relwdom 8747 brwdom 8748 |
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