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Metamath Proof Explorer

Theorem dmmpt

Description: The domain of the mapping operation in general. (Contributed by NM, 16-May-1995) (Revised by Mario Carneiro, 22-Mar-2015)

Ref Expression
Hypothesis dmmpt.1 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
Assertion dmmpt dom 𝐹 = { 𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V }

Proof

Step Hyp Ref Expression
1 dmmpt.1 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
2 dfdm4 ⊢ dom 𝐹 = ran ◡ 𝐹
3 dfrn4 ⊢ ran ◡ 𝐹 = ( ◡ 𝐹 “ V )
4 1 mptpreima ⊢ ( ◡ 𝐹 “ V ) = { 𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V }
5 2 3 4 3eqtri ⊢ dom 𝐹 = { 𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V }