Theorem elpwiuncl 32664

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 4554	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
3	2	ralrimiva 3144	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ⊆ 𝐶)
4		iunss 4992	. . 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 3144	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶)
8	6, 7	jca 518	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶))
9		iunexg 7929	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ V)
10		elpwg 4548	. . 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 398   ∈ wcel 2132  ∀wral 3066  Vcvv 3444   ⊆ wss 3895  𝒫 cpw 4545  ∪ ciun 4939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-11 2181  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1553  df-ex 1790  df-sb 2081  df-mo 2556  df-clab 2731  df-cleq 2744  df-clel 2827  df-ral 3067  df-rex 3077  df-v 3446  df-ss 3912  df-pw 4547  df-uni 4856  df-iun 4941

This theorem is referenced by:  carsggect  34559  carsgclctunlem2  34560