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Metamath Proof Explorer

Theorem rnmptssd

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020)

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

Proof

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