Theorem elpwiuncl 32582

Description: Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.)

Hypotheses

Ref	Expression
elpwiuncl.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
elpwiuncl.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝒫 𝐶)

Assertion

Ref	Expression
elpwiuncl	⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶)

Distinct variable groups:   𝐴,𝑘   𝐶,𝑘   𝜑,𝑘

Allowed substitution hints:   𝐵(𝑘)   𝑉(𝑘)

Proof of Theorem elpwiuncl

Step	Hyp	Ref	Expression
1		elpwiuncl.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝒫 𝐶)
2	1	elpwid 4562	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
3	2	ralrimiva 3127	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶)
4		iunss 4999	. . 3 ⊢ (∪ 𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶)
5	3, 4	sylibr 234	. 2 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶)
6		elpwiuncl.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	1	ralrimiva 3127	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶)
8	6, 7	jca 511	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶))
9		iunexg 7907	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ V)
10		elpwg 4556	. . 3 ⊢ (∪ 𝑘 ∈ 𝐴 𝐵 ∈ V → (∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶 ↔ ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶))
11	8, 9, 10	3syl 18	. 2 ⊢ (𝜑 → (∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶 ↔ ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶))
12	5, 11	mpbird 257	1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∀wral 3050  Vcvv 3439   ⊆ wss 3900  𝒫 cpw 4553  ∪ 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-ext 2707  ax-rep 5223  ax-sep 5240  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-mo 2538  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-v 3441  df-ss 3917  df-pw 4555  df-uni 4863  df-iun 4947

This theorem is referenced by:  carsggect  34454  carsgclctunlem2  34455