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Database SUPPLEMENTARY MATERIAL (USERS' MATHBOXES) Mathbox for Peter Mazsa Preparatory theorems inecmo2
Description: Equivalence of a double restricted universal quantification and a
restricted "at most one" inside a universal quantification.
(Contributed by Peter Mazsa , 29-May-2018) (Revised by Peter Mazsa , 2-Sep-2021)
Ref
Expression
Assertion
inecmo2 ⊢ ( ( ∀ 𝑢 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) ↔ ( ∀ 𝑥 ∃* 𝑢 ∈ 𝐴 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )
Proof
Step
Hyp
Ref
Expression
1
id ⊢ ( 𝑢 = 𝑣 → 𝑢 = 𝑣 )
2
1
inecmo ⊢ ( Rel 𝑅 → ( ∀ 𝑢 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ↔ ∀ 𝑥 ∃* 𝑢 ∈ 𝐴 𝑢 𝑅 𝑥 ) )
3
2
pm5.32ri ⊢ ( ( ∀ 𝑢 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) ↔ ( ∀ 𝑥 ∃* 𝑢 ∈ 𝐴 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )