Theorem elpwiuncl 32732

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 4565	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
3	2	ralrimiva 3155	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶)
4		iunss 5003	. . 3 ⊢ (∪ 𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶)
5	3, 4	sylibr 236	. 2 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶)
6		elpwiuncl.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	1	ralrimiva 3155	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶)
8	6, 7	jca 519	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶))
9		iunexg 7944	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ V)
10		elpwg 4559	. . 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 259	1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2143  ∀wral 3077  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4556  ∪ ciun 4950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1564  df-ex 1801  df-sb 2092  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-v 3457  df-ss 3922  df-pw 4558  df-uni 4867  df-iun 4952

This theorem is referenced by:  carsggect  34617  carsgclctunlem2  34618