Theorem iunssd 5007

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

Hypothesis

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

Assertion

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

Distinct variable groups:   𝑥,𝐶   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem iunssd

Step	Hyp	Ref	Expression
1		iunssd.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
2	1	ralrimiva 3129	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
3		iunss 5001	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4	2, 3	sylibr 234	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3902  ∪ ciun 4947

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-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-v 3443  df-ss 3919  df-iun 4949

This theorem is referenced by:  imasaddfnlem  17453  imasaddflem  17455  subdrgint  20740  bdayiun  27915  precsexlem10  28216  gsumwrd2dccatlem  33161  constrsscn  33899  oacl2g  43639  omcl2  43642  ofoaf  43664  onsucunifi  43679  meaiininclem  46797  smflim  47088  smfresal  47099  smfmullem4  47105  iunlub  49133