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Theorem elgrug 9493
 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 4686 . . 3 (𝑢 = 𝑈 → (Tr 𝑢 ↔ Tr 𝑈))
2 eleq2 2677 . . . . 5 (𝑢 = 𝑈 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑥𝑈))
3 eleq2 2677 . . . . . 6 (𝑢 = 𝑈 → ({𝑥, 𝑦} ∈ 𝑢 ↔ {𝑥, 𝑦} ∈ 𝑈))
43raleqbi1dv 3123 . . . . 5 (𝑢 = 𝑈 → (∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ↔ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
5 oveq1 6556 . . . . . 6 (𝑢 = 𝑈 → (𝑢𝑚 𝑥) = (𝑈𝑚 𝑥))
6 eleq2 2677 . . . . . 6 (𝑢 = 𝑈 → ( ran 𝑦𝑢 ran 𝑦𝑈))
75, 6raleqbidv 3129 . . . . 5 (𝑢 = 𝑈 → (∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢 ↔ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈))
82, 4, 73anbi123d 1391 . . . 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 3123 . . 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 743 . 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 9492 . 2 Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢))}
1210, 11elab2g 3322 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  𝒫 cpw 4108  {cpr 4127  ∪ cuni 4372  Tr wtr 4680  ran crn 5039  (class class class)co 6549   ↑𝑚 cmap 7744  Univcgru 9491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-tr 4681  df-iota 5768  df-fv 5812  df-ov 6552  df-gru 9492 This theorem is referenced by:  grutr  9494  grupw  9496  grupr  9498  gruurn  9499  intgru  9515  ingru  9516  grutsk1  9522
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