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Theorem grupw 10013
 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 10010 . . . . 5 (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈𝑚 𝑦) ran 𝑥𝑈))))
21ibi 259 . . . 4 (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈𝑚 𝑦) ran 𝑥𝑈)))
32simprd 488 . . 3 (𝑈 ∈ Univ → ∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈𝑚 𝑦) ran 𝑥𝑈))
4 simp1 1117 . . . 4 ((𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈𝑚 𝑦) ran 𝑥𝑈) → 𝒫 𝑦𝑈)
54ralimi 3103 . . 3 (∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈𝑚 𝑦) ran 𝑥𝑈) → ∀𝑦𝑈 𝒫 𝑦𝑈)
6 pweq 4419 . . . . 5 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
76eleq1d 2843 . . . 4 (𝑦 = 𝐴 → (𝒫 𝑦𝑈 ↔ 𝒫 𝐴𝑈))
87rspccv 3525 . . 3 (∀𝑦𝑈 𝒫 𝑦𝑈 → (𝐴𝑈 → 𝒫 𝐴𝑈))
93, 5, 83syl 18 . 2 (𝑈 ∈ Univ → (𝐴𝑈 → 𝒫 𝐴𝑈))
109imp 398 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝒫 𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051  ∀wral 3081  𝒫 cpw 4416  {cpr 4437  ∪ cuni 4708  Tr wtr 5026  ran crn 5404  (class class class)co 6974   ↑𝑚 cmap 8204  Univcgru 10008 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-tr 5027  df-iota 6149  df-fv 6193  df-ov 6977  df-gru 10009 This theorem is referenced by:  gruss  10014  grurn  10019  gruxp  10025  grumap  10026  gruwun  10031  intgru  10032  gruina  10036  grur1a  10037  grur1cld  39981  grumnudlem  40034
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