Axiom ax-dc

Description: Dependent Choice. Axiom DC1 of Schechter p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. Dependent choice is equivalent to the statement that every (nonempty) pruned tree has a branch. This axiom is redundant in ZFC; see axdc . But ZF+DC is strictly weaker than ZF+AC, so this axiom provides for theorems that do not need the full power of AC. (Contributed by Mario Carneiro, 25-Jan-2013)

	Ref	Expression
Assertion	ax-dc	⊢ ( ( ∃ 𝑦 ∃ 𝑧 𝑦 𝑥 𝑧 ∧ ran 𝑥 ⊆ dom 𝑥 ) → ∃ 𝑓 ∀ 𝑛 ∈ ω ( 𝑓 ‘ 𝑛 ) 𝑥 ( 𝑓 ‘ suc 𝑛 ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		vy	⊢ 𝑦
1		vz	⊢ 𝑧
2	0	cv	⊢ 𝑦
3		vx	⊢ 𝑥
4	3	cv	⊢ 𝑥
5	1	cv	⊢ 𝑧
6	2 5 4	wbr	⊢ 𝑦 𝑥 𝑧
7	6 1	wex	⊢ ∃ 𝑧 𝑦 𝑥 𝑧
8	7 0	wex	⊢ ∃ 𝑦 ∃ 𝑧 𝑦 𝑥 𝑧
9	4	crn	⊢ ran 𝑥
10	4	cdm	⊢ dom 𝑥
11	9 10	wss	⊢ ran 𝑥 ⊆ dom 𝑥
12	8 11	wa	⊢ ( ∃ 𝑦 ∃ 𝑧 𝑦 𝑥 𝑧 ∧ ran 𝑥 ⊆ dom 𝑥 )
13		vf	⊢ 𝑓
14		vn	⊢ 𝑛
15		com	⊢ ω
16	13	cv	⊢ 𝑓
17	14	cv	⊢ 𝑛
18	17 16	cfv	⊢ ( 𝑓 ‘ 𝑛 )
19	17	csuc	⊢ suc 𝑛
20	19 16	cfv	⊢ ( 𝑓 ‘ suc 𝑛 )
21	18 20 4	wbr	⊢ ( 𝑓 ‘ 𝑛 ) 𝑥 ( 𝑓 ‘ suc 𝑛 )
22	21 14 15	wral	⊢ ∀ 𝑛 ∈ ω ( 𝑓 ‘ 𝑛 ) 𝑥 ( 𝑓 ‘ suc 𝑛 )
23	22 13	wex	⊢ ∃ 𝑓 ∀ 𝑛 ∈ ω ( 𝑓 ‘ 𝑛 ) 𝑥 ( 𝑓 ‘ suc 𝑛 )
24	12 23	wi	⊢ ( ( ∃ 𝑦 ∃ 𝑧 𝑦 𝑥 𝑧 ∧ ran 𝑥 ⊆ dom 𝑥 ) → ∃ 𝑓 ∀ 𝑛 ∈ ω ( 𝑓 ‘ 𝑛 ) 𝑥 ( 𝑓 ‘ suc 𝑛 ) )