Theorem grupw 10482

Description: A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
grupw	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈)

Proof of Theorem grupw

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elgrug 10479	. . . . 5 ⊢ (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈))))
2	1	ibi 266	. . . 4 ⊢ (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈)))
3	2	simprd 495	. . 3 ⊢ (𝑈 ∈ Univ → ∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈))
4		simp1 1134	. . . 4 ⊢ ((𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈) → 𝒫 𝑦 ∈ 𝑈)
5	4	ralimi 3086	. . 3 ⊢ (∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈) → ∀𝑦 ∈ 𝑈 𝒫 𝑦 ∈ 𝑈)
6		pweq 4546	. . . . 5 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
7	6	eleq1d 2823	. . . 4 ⊢ (𝑦 = 𝐴 → (𝒫 𝑦 ∈ 𝑈 ↔ 𝒫 𝐴 ∈ 𝑈))
8	7	rspccv 3549	. . 3 ⊢ (∀𝑦 ∈ 𝑈 𝒫 𝑦 ∈ 𝑈 → (𝐴 ∈ 𝑈 → 𝒫 𝐴 ∈ 𝑈))
9	3, 5, 8	3syl 18	. 2 ⊢ (𝑈 ∈ Univ → (𝐴 ∈ 𝑈 → 𝒫 𝐴 ∈ 𝑈))
10	9	imp 406	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  𝒫 cpw 4530  {cpr 4560  ∪ cuni 4836  Tr wtr 5187  ran crn 5581  (class class class)co 7255   ↑_m cmap 8573  Univcgru 10477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-tr 5188  df-iota 6376  df-fv 6426  df-ov 7258  df-gru 10478

This theorem is referenced by:  gruss  10483  grurn  10488  gruxp  10494  grumap  10495  gruwun  10500  intgru  10501  gruina  10505  grur1a  10506  grur1cld  41739  grumnudlem  41792