Theorem ineccnvmo

Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021)

	Ref	Expression
Assertion	ineccnvmo	⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑦 = 𝑧 ∨ ( [ 𝑦 ] ◡ 𝐹 ∩ [ 𝑧 ] ◡ 𝐹 ) = ∅ ) ↔ ∀ 𝑥 ∃* 𝑦 ∈ 𝐵 𝑥 𝐹 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		relcnv	⊢ Rel ◡ 𝐹
2		id	⊢ ( 𝑦 = 𝑧 → 𝑦 = 𝑧 )
3	2	inecmo	⊢ ( Rel ◡ 𝐹 → ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑦 = 𝑧 ∨ ( [ 𝑦 ] ◡ 𝐹 ∩ [ 𝑧 ] ◡ 𝐹 ) = ∅ ) ↔ ∀ 𝑥 ∃* 𝑦 ∈ 𝐵 𝑦 ◡ 𝐹 𝑥 ) )
4	1 3	ax-mp	⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑦 = 𝑧 ∨ ( [ 𝑦 ] ◡ 𝐹 ∩ [ 𝑧 ] ◡ 𝐹 ) = ∅ ) ↔ ∀ 𝑥 ∃* 𝑦 ∈ 𝐵 𝑦 ◡ 𝐹 𝑥 )
5		brcnvg	⊢ ( ( 𝑦 ∈ V ∧ 𝑥 ∈ V ) → ( 𝑦 ◡ 𝐹 𝑥 ↔ 𝑥 𝐹 𝑦 ) )
6	5	el2v	⊢ ( 𝑦 ◡ 𝐹 𝑥 ↔ 𝑥 𝐹 𝑦 )
7	6	rmobii	⊢ ( ∃* 𝑦 ∈ 𝐵 𝑦 ◡ 𝐹 𝑥 ↔ ∃* 𝑦 ∈ 𝐵 𝑥 𝐹 𝑦 )
8	7	albii	⊢ ( ∀ 𝑥 ∃* 𝑦 ∈ 𝐵 𝑦 ◡ 𝐹 𝑥 ↔ ∀ 𝑥 ∃* 𝑦 ∈ 𝐵 𝑥 𝐹 𝑦 )
9	4 8	bitri	⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑦 = 𝑧 ∨ ( [ 𝑦 ] ◡ 𝐹 ∩ [ 𝑧 ] ◡ 𝐹 ) = ∅ ) ↔ ∀ 𝑥 ∃* 𝑦 ∈ 𝐵 𝑥 𝐹 𝑦 )