This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem funcnv

Description: The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that x A y . Definition of single-rooted in Enderton p. 43. See funcnv2 for a simpler version. (Contributed by NM, 13-Aug-2004)

Ref Expression
Assertion funcnv ( Fun ◡ 𝐴 ↔ ∀ 𝑦 ∈ ran 𝐴 ∃* 𝑥 𝑥 𝐴 𝑦 )

Proof

Step Hyp Ref Expression
1 vex ⊢ 𝑥 ∈ V
2 vex ⊢ 𝑦 ∈ V
3 1 2 brelrn ⊢ ( 𝑥 𝐴 𝑦 → 𝑦 ∈ ran 𝐴 )
4 3 pm4.71ri ⊢ ( 𝑥 𝐴 𝑦 ↔ ( 𝑦 ∈ ran 𝐴 ∧ 𝑥 𝐴 𝑦 ) )
5 4 mobii ⊢ ( ∃* 𝑥 𝑥 𝐴 𝑦 ↔ ∃* 𝑥 ( 𝑦 ∈ ran 𝐴 ∧ 𝑥 𝐴 𝑦 ) )
6 moanimv ⊢ ( ∃* 𝑥 ( 𝑦 ∈ ran 𝐴 ∧ 𝑥 𝐴 𝑦 ) ↔ ( 𝑦 ∈ ran 𝐴 → ∃* 𝑥 𝑥 𝐴 𝑦 ) )
7 5 6 bitri ⊢ ( ∃* 𝑥 𝑥 𝐴 𝑦 ↔ ( 𝑦 ∈ ran 𝐴 → ∃* 𝑥 𝑥 𝐴 𝑦 ) )
8 7 albii ⊢ ( ∀ 𝑦 ∃* 𝑥 𝑥 𝐴 𝑦 ↔ ∀ 𝑦 ( 𝑦 ∈ ran 𝐴 → ∃* 𝑥 𝑥 𝐴 𝑦 ) )
9 funcnv2 ⊢ ( Fun ◡ 𝐴 ↔ ∀ 𝑦 ∃* 𝑥 𝑥 𝐴 𝑦 )
10 df-ral ⊢ ( ∀ 𝑦 ∈ ran 𝐴 ∃* 𝑥 𝑥 𝐴 𝑦 ↔ ∀ 𝑦 ( 𝑦 ∈ ran 𝐴 → ∃* 𝑥 𝑥 𝐴 𝑦 ) )
11 8 9 10 3bitr4i ⊢ ( Fun ◡ 𝐴 ↔ ∀ 𝑦 ∈ ran 𝐴 ∃* 𝑥 𝑥 𝐴 𝑦 )