Theorem ineccnvmo2

Description: Equivalence of a double universal quantification restricted to the range and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 4-Sep-2021)

	Ref	Expression
Assertion	ineccnvmo2	⊢ ( ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝐹 ∩ [ 𝑦 ] ◡ 𝐹 ) = ∅ ) ↔ ∀ 𝑢 ∃* 𝑥 𝑢 𝐹 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		ineccnvmo	⊢ ( ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝐹 ∩ [ 𝑦 ] ◡ 𝐹 ) = ∅ ) ↔ ∀ 𝑢 ∃* 𝑥 ∈ ran 𝐹 𝑢 𝐹 𝑥 )
2		alrmomorn	⊢ ( ∀ 𝑢 ∃* 𝑥 ∈ ran 𝐹 𝑢 𝐹 𝑥 ↔ ∀ 𝑢 ∃* 𝑥 𝑢 𝐹 𝑥 )
3	1 2	bitri	⊢ ( ∀ 𝑥 ∈ ran 𝐹 ∀ 𝑦 ∈ ran 𝐹 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝐹 ∩ [ 𝑦 ] ◡ 𝐹 ) = ∅ ) ↔ ∀ 𝑢 ∃* 𝑥 𝑢 𝐹 𝑥 )