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Theorem elgrug 9902
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 4951 . . 3 (𝑢 = 𝑈 → (Tr 𝑢 ↔ Tr 𝑈))
2 eleq2 2867 . . . . 5 (𝑢 = 𝑈 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑥𝑈))
3 eleq2 2867 . . . . . 6 (𝑢 = 𝑈 → ({𝑥, 𝑦} ∈ 𝑢 ↔ {𝑥, 𝑦} ∈ 𝑈))
43raleqbi1dv 3329 . . . . 5 (𝑢 = 𝑈 → (∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ↔ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
5 oveq1 6885 . . . . . 6 (𝑢 = 𝑈 → (𝑢𝑚 𝑥) = (𝑈𝑚 𝑥))
6 eleq2 2867 . . . . . 6 (𝑢 = 𝑈 → ( ran 𝑦𝑢 ran 𝑦𝑈))
75, 6raleqbidv 3335 . . . . 5 (𝑢 = 𝑈 → (∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢 ↔ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈))
82, 4, 73anbi123d 1561 . . . 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 3329 . . 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 625 . 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 9901 . 2 Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢))}
1210, 11elab2g 3545 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 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  𝒫 cpw 4349  {cpr 4370   cuni 4628  Tr wtr 4945  ran crn 5313  (class class class)co 6878  𝑚 cmap 8095  Univcgru 9900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-tr 4946  df-iota 6064  df-fv 6109  df-ov 6881  df-gru 9901
This theorem is referenced by:  grutr  9903  grupw  9905  grupr  9907  gruurn  9908  intgru  9924  ingru  9925  grutsk1  9931
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