MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grupw Structured version   Visualization version   GIF version

Theorem grupw 10812
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 10809 . . . . 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 1134 . . . 4 ((𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈) → 𝒫 𝑦𝑈)
54ralimi 3078 . . 3 (∀𝑦𝑈 (𝒫 𝑦𝑈 ∧ ∀𝑥𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈m 𝑦) ran 𝑥𝑈) → ∀𝑦𝑈 𝒫 𝑦𝑈)
6 pweq 4612 . . . . 5 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
76eleq1d 2813 . . . 4 (𝑦 = 𝐴 → (𝒫 𝑦𝑈 ↔ 𝒫 𝐴𝑈))
87rspccv 3604 . . 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 1085   = wceq 1534  wcel 2099  wral 3056  𝒫 cpw 4598  {cpr 4626   cuni 4903  Tr wtr 5259  ran crn 5673  (class class class)co 7414  m cmap 8838  Univcgru 10807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-tr 5260  df-iota 6494  df-fv 6550  df-ov 7417  df-gru 10808
This theorem is referenced by:  gruss  10813  grurn  10818  gruxp  10824  grumap  10825  gruwun  10830  intgru  10831  gruina  10835  grur1a  10836  grur1cld  43641  grumnudlem  43694
  Copyright terms: Public domain W3C validator