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Metamath Proof Explorer

Theorem iunssd

Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021)

Ref Expression
Hypothesis iunssd.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ⊆ 𝐶 )
Assertion iunssd ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )

Proof

Step Hyp Ref Expression
1 iunssd.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ⊆ 𝐶 )
2 1 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
3 iunss ⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
4 2 3 sylibr ⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )