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Metamath Proof Explorer

Theorem iunssdf

Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 24-Jan-2025)

Ref Expression
Hypotheses iunssdf.1 ⊢ Ⅎ 𝑥 𝜑
iunssdf.2 ⊢ Ⅎ 𝑥 𝐶
iunssdf.3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ⊆ 𝐶 )
Assertion iunssdf ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )

Proof

Step Hyp Ref Expression
1 iunssdf.1 ⊢ Ⅎ 𝑥 𝜑
2 iunssdf.2 ⊢ Ⅎ 𝑥 𝐶
3 iunssdf.3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ⊆ 𝐶 )
4 1 3 ralrimia ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
5 2 iunssf ⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
6 4 5 sylibr ⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )