MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-wdom Structured version   Visualization version   GIF version

Definition df-wdom 9324
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 8735) implies weak dominance (over ZF). The principle asserting the converse is known as the partition principle and is independent of ZF. Theorem fodom 10279 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 9323 . 2 class *
2 vx . . . . . 6 setvar 𝑥
32cv 1538 . . . . 5 class 𝑥
4 c0 4256 . . . . 5 class
53, 4wceq 1539 . . . 4 wff 𝑥 = ∅
6 vy . . . . . . 7 setvar 𝑦
76cv 1538 . . . . . 6 class 𝑦
8 vz . . . . . . 7 setvar 𝑧
98cv 1538 . . . . . 6 class 𝑧
107, 3, 9wfo 6431 . . . . 5 wff 𝑧:𝑦onto𝑥
1110, 8wex 1782 . . . 4 wff 𝑧 𝑧:𝑦onto𝑥
125, 11wo 844 . . 3 wff (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)
1312, 2, 6copab 5136 . 2 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
141, 13wceq 1539 1 wff * = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
Colors of variables: wff setvar class
This definition is referenced by:  relwdom  9325  brwdom  9326
  Copyright terms: Public domain W3C validator