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Metamath Proof Explorer

Theorem rnmptssdff

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 24-Jan-2025)

Ref Expression
Hypotheses rnmptssdff.1 ⊢ Ⅎ 𝑥 𝜑
rnmptssdff.2 ⊢ Ⅎ 𝑥 𝐴
rnmptssdff.3 ⊢ Ⅎ 𝑥 𝐶
rnmptssdff.4 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
rnmptssdff.5 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
Assertion rnmptssdff ( 𝜑 → ran 𝐹 ⊆ 𝐶 )

Proof

Step Hyp Ref Expression
1 rnmptssdff.1 ⊢ Ⅎ 𝑥 𝜑
2 rnmptssdff.2 ⊢ Ⅎ 𝑥 𝐴
3 rnmptssdff.3 ⊢ Ⅎ 𝑥 𝐶
4 rnmptssdff.4 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
5 rnmptssdff.5 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
6 1 5 ralrimia ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 )
7 2 3 4 rnmptssff ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶 )
8 6 7 syl ⊢ ( 𝜑 → ran 𝐹 ⊆ 𝐶 )