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Theorem elgrug 10686
Description: Properties of a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
elgrug (𝑈𝑉 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))))
Distinct variable group:   𝑥,𝑈,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem elgrug
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 treq 5228 . . 3 (𝑢 = 𝑈 → (Tr 𝑢 ↔ Tr 𝑈))
2 eleq2 2826 . . . . 5 (𝑢 = 𝑈 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑥𝑈))
3 eleq2 2826 . . . . . 6 (𝑢 = 𝑈 → ({𝑥, 𝑦} ∈ 𝑢 ↔ {𝑥, 𝑦} ∈ 𝑈))
43raleqbi1dv 3305 . . . . 5 (𝑢 = 𝑈 → (∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ↔ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
5 oveq1 7358 . . . . . 6 (𝑢 = 𝑈 → (𝑢m 𝑥) = (𝑈m 𝑥))
6 eleq2 2826 . . . . . 6 (𝑢 = 𝑈 → ( ran 𝑦𝑢 ran 𝑦𝑈))
75, 6raleqbidv 3317 . . . . 5 (𝑢 = 𝑈 → (∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢 ↔ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
82, 4, 73anbi123d 1436 . . . 4 (𝑢 = 𝑈 → ((𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢) ↔ (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈)))
98raleqbi1dv 3305 . . 3 (𝑢 = 𝑈 → (∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢) ↔ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈)))
101, 9anbi12d 631 . 2 (𝑢 = 𝑈 → ((Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢)) ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))))
11 df-gru 10685 . 2 Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢))}
1210, 11elab2g 3630 1 (𝑈𝑉 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3062  𝒫 cpw 4558  {cpr 4586   cuni 4863  Tr wtr 5220  ran crn 5632  (class class class)co 7351  m cmap 8723  Univcgru 10684
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-tr 5221  df-iota 6445  df-fv 6501  df-ov 7354  df-gru 10685
This theorem is referenced by:  grutr  10687  grupw  10689  grupr  10691  gruurn  10692  intgru  10708  ingru  10709  grutsk1  10715  mnugrud  42475
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