Metamath Proof Explorer

Theorem dmafv2rnb

Description: The alternate function value at a class A is defined, i.e., in the range of the function, iff A is in the domain of the function. (Contributed by AV, 3-Sep-2022)

Ref Expression
Assertion dmafv2rnb ⊢ Fun ⁡ F ↾ A → A ∈ dom ⁡ F ↔ F '''' A ∈ ran ⁡ F

Proof

Step Hyp Ref Expression
1 iba ⊢ Fun ⁡ F ↾ A → A ∈ dom ⁡ F ↔ A ∈ dom ⁡ F ∧ Fun ⁡ F ↾ A
2 df-dfat ⊢ F defAt A ↔ A ∈ dom ⁡ F ∧ Fun ⁡ F ↾ A
3 dfatafv2rnb ⊢ F defAt A ↔ F '''' A ∈ ran ⁡ F
4 2 3 bitr3i ⊢ A ∈ dom ⁡ F ∧ Fun ⁡ F ↾ A ↔ F '''' A ∈ ran ⁡ F
5 1 4 syl6bb ⊢ Fun ⁡ F ↾ A → A ∈ dom ⁡ F ↔ F '''' A ∈ ran ⁡ F