Theorem inecmo3

Description: Equivalence of a double universal quantification restricted to the domain and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	inecmo3	⊢ ( ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) ↔ ( ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		inecmo2	⊢ ( ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) ↔ ( ∀ 𝑥 ∃* 𝑢 ∈ dom 𝑅 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )
2		alrmomodm	⊢ ( Rel 𝑅 → ( ∀ 𝑥 ∃* 𝑢 ∈ dom 𝑅 𝑢 𝑅 𝑥 ↔ ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ) )
3	2	pm5.32ri	⊢ ( ( ∀ 𝑥 ∃* 𝑢 ∈ dom 𝑅 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) ↔ ( ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )
4	1 3	bitri	⊢ ( ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) ↔ ( ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )