Theorem elpwunicl 29371

Description: Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Hypotheses

Ref	Expression
elpwunicl.1	⊢ (𝜑 → 𝐵 ∈ 𝑉)
elpwunicl.2	⊢ (𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵)

Assertion

Ref	Expression
elpwunicl	⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwunicl

Step	Hyp	Ref	Expression
1		elpwunicl.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵)
2		elpwg 4166	. . . . 5 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵))
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵))
4	1, 3	mpbid 222	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝐵)
5		pwuniss 29370	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ 𝐴 ⊆ 𝐵)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → ∪ 𝐴 ⊆ 𝐵)
7		uniexg 6955	. . 3 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 → ∪ 𝐴 ∈ V)
8		elpwg 4166	. . 3 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
9	1, 7, 8	3syl 18	. 2 ⊢ (𝜑 → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
10	6, 9	mpbird 247	1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  ldgenpisyslem1  30226