Metamath Proof Explorer


Theorem elrnmptd

Description: The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses elrnmptd.f 𝐹 = ( 𝑥𝐴𝐵 )
elrnmptd.x ( 𝜑 → ∃ 𝑥𝐴 𝐶 = 𝐵 )
elrnmptd.c ( 𝜑𝐶𝑉 )
Assertion elrnmptd ( 𝜑𝐶 ∈ ran 𝐹 )

Proof

Step Hyp Ref Expression
1 elrnmptd.f 𝐹 = ( 𝑥𝐴𝐵 )
2 elrnmptd.x ( 𝜑 → ∃ 𝑥𝐴 𝐶 = 𝐵 )
3 elrnmptd.c ( 𝜑𝐶𝑉 )
4 1 elrnmpt ( 𝐶𝑉 → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥𝐴 𝐶 = 𝐵 ) )
5 3 4 syl ( 𝜑 → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥𝐴 𝐶 = 𝐵 ) )
6 2 5 mpbird ( 𝜑𝐶 ∈ ran 𝐹 )