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Theorem grupw 10833
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 ∧ 𝐴𝑈) → 𝒫 𝐴𝑈)

Proof of Theorem grupw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elgrug 10830 . . . . 5 (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈))))
21ibi 267 . . . 4 (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈)))
32simprd 495 . . 3 (𝑈 ∈ Univ → ∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈))
4 simp1 1135 . . . 4 ((𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈) → 𝒫 𝑦𝑈)
54ralimi 3081 . . 3 (∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈) → ∀𝑦𝑈 𝒫 𝑦𝑈)
6 pweq 4619 . . . . 5 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
76eleq1d 2824 . . . 4 (𝑦 = 𝐴 → (𝒫 𝑦𝑈 ↔ 𝒫 𝐴𝑈))
87rspccv 3619 . . 3 (∀𝑦𝑈 𝒫 𝑦𝑈 → (𝐴𝑈 → 𝒫 𝐴𝑈))
93, 5, 83syl 18 . 2 (𝑈 ∈ Univ → (𝐴𝑈 → 𝒫 𝐴𝑈))
109imp 406 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝒫 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  𝒫 cpw 4605  {cpr 4633   cuni 4912  Tr wtr 5265  ran crn 5690  (class class class)co 7431  m cmap 8865  Univcgru 10828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-tr 5266  df-iota 6516  df-fv 6571  df-ov 7434  df-gru 10829
This theorem is referenced by:  gruss  10834  grurn  10839  gruxp  10845  grumap  10846  gruwun  10851  intgru  10852  gruina  10856  grur1a  10857  grur1cld  44228  grumnudlem  44281
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