Metamath Proof Explorer

Theorem foeq2

Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994)

Ref Expression
Assertion foeq2 ⊢ A = B → F : A ⟶ onto C ↔ F : B ⟶ onto C

Proof

Step Hyp Ref Expression
1 fneq2 ⊢ A = B → F Fn A ↔ F Fn B
2 1 anbi1d ⊢ A = B → F Fn A ∧ ran ⁡ F = C ↔ F Fn B ∧ ran ⁡ F = C
3 df-fo ⊢ F : A ⟶ onto C ↔ F Fn A ∧ ran ⁡ F = C
4 df-fo ⊢ F : B ⟶ onto C ↔ F Fn B ∧ ran ⁡ F = C
5 2 3 4 3bitr4g ⊢ A = B → F : A ⟶ onto C ↔ F : B ⟶ onto C