Theorem iunssdf 45437

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

Hypotheses

Ref	Expression
iunssdf.1	⊢ Ⅎ𝑥𝜑
iunssdf.2	⊢ Ⅎ𝑥𝐶
iunssdf.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
iunssdf	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

Proof of Theorem iunssdf

Step	Hyp	Ref	Expression
1		iunssdf.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		iunssdf.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
3	1, 2	ralrimia 3234	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4		iunssdf.2	. . 3 ⊢ Ⅎ𝑥𝐶
5	4	iunssf 4997	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
6	3, 5	sylibr 234	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2882  ∀wral 3050   ⊆ wss 3900  ∪ ciun 4945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2183  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3051  df-rex 3060  df-v 3441  df-ss 3917  df-iun 4947

This theorem is referenced by:  fsupdm  47123  finfdm  47127