Metamath Proof Explorer

Theorem dmmptg

Description: The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013) (Revised by Mario Carneiro, 14-Sep-2013)

Ref Expression
Assertion dmmptg ⊢ ∀ x ∈ A B ∈ V → dom ⁡ x ∈ A ⟼ B = A

Proof

Step Hyp Ref Expression
1 elex ⊢ B ∈ V → B ∈ V
2 1 ralimi ⊢ ∀ x ∈ A B ∈ V → ∀ x ∈ A B ∈ V
3 rabid2 ⊢ A = x ∈ A | B ∈ V ↔ ∀ x ∈ A B ∈ V
4 2 3 sylibr ⊢ ∀ x ∈ A B ∈ V → A = x ∈ A | B ∈ V
5 eqid ⊢ x ∈ A ⟼ B = x ∈ A ⟼ B
6 5 dmmpt ⊢ dom ⁡ x ∈ A ⟼ B = x ∈ A | B ∈ V
7 4 6 syl6reqr ⊢ ∀ x ∈ A B ∈ V → dom ⁡ x ∈ A ⟼ B = A