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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 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
elrnmptd.x ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 )
elrnmptd.c ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
Assertion elrnmptd ( 𝜑 → 𝐶 ∈ ran 𝐹 )

Proof

Step Hyp Ref Expression
1 elrnmptd.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
2 elrnmptd.x ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 )
3 elrnmptd.c ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
4 1 elrnmpt ⊢ ( 𝐶 ∈ 𝑉 → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) )
5 3 4 syl ⊢ ( 𝜑 → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) )
6 2 5 mpbird ⊢ ( 𝜑 → 𝐶 ∈ ran 𝐹 )