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Metamath Proof Explorer

Theorem rnmptssdf

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses rnmptssdf.1 ⊢ Ⅎ 𝑥 𝜑
rnmptssdf.2 ⊢ Ⅎ 𝑥 𝐶
rnmptssdf.3 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
rnmptssdf.4 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
Assertion rnmptssdf ( 𝜑 → ran 𝐹 ⊆ 𝐶 )

Proof

Step Hyp Ref Expression
1 rnmptssdf.1 ⊢ Ⅎ 𝑥 𝜑
2 rnmptssdf.2 ⊢ Ⅎ 𝑥 𝐶
3 rnmptssdf.3 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
4 rnmptssdf.4 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
5 1 4 ralrimia ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 )
6 2 3 rnmptssf ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶 )
7 5 6 syl ⊢ ( 𝜑 → ran 𝐹 ⊆ 𝐶 )