Theorem iunssd 4982

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 3127	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
3		iunss 4976	. 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 3049   ⊆ wss 3885  ∪ ciun 4923

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3050  df-rex 3060  df-v 3429  df-ss 3902  df-iun 4925

This theorem is referenced by:  imasaddfnlem  17481  imasaddflem  17483  subdrgint  20769  bdayiun  27895  precsexlem10  28196  gsumwrd2dccatlem  33126  constrsscn  33872  ttcmin  36666  dfttc2g  36676  oacl2g  43746  omcl2  43749  ofoaf  43771  onsucunifi  43786  meaiininclem  46902  smflim  47193  smfresal  47204  smfmullem4  47210  iunlub  49284