Description: Equivalent of Axiom of Choice. B is a collection B ( x ) of nonempty sets. Remark after Theorem 10.46 of TakeutiZaring p. 98. (Contributed by Mario Carneiro, 22-Mar-2013)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ac6c4.1 | ⊢ 𝐴 ∈ V | |
ac6c4.2 | ⊢ 𝐵 ∈ V | ||
Assertion | ac6c5 | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃ 𝑓 ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ac6c4.1 | ⊢ 𝐴 ∈ V | |
2 | ac6c4.2 | ⊢ 𝐵 ∈ V | |
3 | 1 2 | ac6c4 | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) ) |
4 | exsimpr | ⊢ ( ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) → ∃ 𝑓 ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) | |
5 | 3 4 | syl | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃ 𝑓 ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) |