Metamath Proof Explorer

Theorem f1eq3

Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997)

Ref Expression
Assertion f1eq3 ⊢ A = B → F : C ⟶ 1-1 A ↔ F : C ⟶ 1-1 B

Proof

Step Hyp Ref Expression
1 feq3 ⊢ A = B → F : C ⟶ A ↔ F : C ⟶ B
2 1 anbi1d ⊢ A = B → F : C ⟶ A ∧ Fun ⁡ F -1 ↔ F : C ⟶ B ∧ Fun ⁡ F -1
3 df-f1 ⊢ F : C ⟶ 1-1 A ↔ F : C ⟶ A ∧ Fun ⁡ F -1
4 df-f1 ⊢ F : C ⟶ 1-1 B ↔ F : C ⟶ B ∧ Fun ⁡ F -1
5 2 3 4 3bitr4g ⊢ A = B → F : C ⟶ 1-1 A ↔ F : C ⟶ 1-1 B