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Metamath Proof Explorer

Theorem opelcnvg

Description: Ordered-pair membership in converse relation. (Contributed by NM, 13-May-1999) (Proof shortened by Andrew Salmon, 27-Aug-2011)

Ref Expression
Assertion opelcnvg ⊢ A ∈ C ∧ B ∈ D → A B ∈ R -1 ↔ B A ∈ R

Proof

Step Hyp Ref Expression
1 brcnvg ⊢ A ∈ C ∧ B ∈ D → A R -1 B ↔ B R A
2 df-br ⊢ A R -1 B ↔ A B ∈ R -1
3 df-br ⊢ B R A ↔ B A ∈ R
4 1 2 3 3bitr3g ⊢ A ∈ C ∧ B ∈ D → A B ∈ R -1 ↔ B A ∈ R