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Theorem grupw 10210
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 10207 . . . . 5 (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈))))
21ibi 270 . . . 4 (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈)))
32simprd 499 . . 3 (𝑈 ∈ Univ → ∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈))
4 simp1 1133 . . . 4 ((𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈) → 𝒫 𝑦𝑈)
54ralimi 3131 . . 3 (∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈) → ∀𝑦𝑈 𝒫 𝑦𝑈)
6 pweq 4516 . . . . 5 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
76eleq1d 2877 . . . 4 (𝑦 = 𝐴 → (𝒫 𝑦𝑈 ↔ 𝒫 𝐴𝑈))
87rspccv 3571 . . 3 (∀𝑦𝑈 𝒫 𝑦𝑈 → (𝐴𝑈 → 𝒫 𝐴𝑈))
93, 5, 83syl 18 . 2 (𝑈 ∈ Univ → (𝐴𝑈 → 𝒫 𝐴𝑈))
109imp 410 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝒫 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  𝒫 cpw 4500  {cpr 4530   cuni 4803  Tr wtr 5139  ran crn 5524  (class class class)co 7139  m cmap 8393  Univcgru 10205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-tr 5140  df-iota 6287  df-fv 6336  df-ov 7142  df-gru 10206
This theorem is referenced by:  gruss  10211  grurn  10216  gruxp  10222  grumap  10223  gruwun  10228  intgru  10229  gruina  10233  grur1a  10234  grur1cld  40937  grumnudlem  40990
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