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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 ( ∀ 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 )