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Theorem grupw 10835
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 10832 . . . . 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 1137 . . . 4 ((𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈) → 𝒫 𝑦𝑈)
54ralimi 3083 . . 3 (∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈) → ∀𝑦𝑈 𝒫 𝑦𝑈)
6 pweq 4614 . . . . 5 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
76eleq1d 2826 . . . 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 1087   = wceq 1540  wcel 2108  wral 3061  𝒫 cpw 4600  {cpr 4628   cuni 4907  Tr wtr 5259  ran crn 5686  (class class class)co 7431  m cmap 8866  Univcgru 10830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-tr 5260  df-iota 6514  df-fv 6569  df-ov 7434  df-gru 10831
This theorem is referenced by:  gruss  10836  grurn  10841  gruxp  10847  grumap  10848  gruwun  10853  intgru  10854  gruina  10858  grur1a  10859  grur1cld  44251  grumnudlem  44304
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