Theorem iunssdf 45160

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 3245	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4		iunssdf.2	. . 3 ⊢ Ⅎ𝑥𝐶
5	4	iunssf 5025	. 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 1783   ∈ wcel 2109  Ⅎwnfc 2884  ∀wral 3052   ⊆ wss 3931  ∪ ciun 4972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-ss 3948  df-iun 4974

This theorem is referenced by:  fsupdm  46851  finfdm  46855