Metamath Proof Explorer


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 ( ∀ 𝑦𝐵𝑧𝐵 ( 𝑦 = 𝑧 ∨ ( [ 𝑦 ] 𝐹 ∩ [ 𝑧 ] 𝐹 ) = ∅ ) ↔ ∀ 𝑥 ∃* 𝑦𝐵 𝑥 𝐹 𝑦 )

Proof

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