Theorem elgrug 10203

Description: Properties of a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
elgrug	⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))))

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.

Step	Hyp	Ref	Expression
1		treq 5142	. . 3 ⊢ (𝑢 = 𝑈 → (Tr 𝑢 ↔ Tr 𝑈))
2		eleq2 2878	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝒫 𝑥 ∈ 𝑢 ↔ 𝒫 𝑥 ∈ 𝑈))
3		eleq2 2878	. . . . . 6 ⊢ (𝑢 = 𝑈 → ({𝑥, 𝑦} ∈ 𝑢 ↔ {𝑥, 𝑦} ∈ 𝑈))
4	3	raleqbi1dv 3356	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ↔ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
5		oveq1 7142	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 ↑m 𝑥) = (𝑈 ↑m 𝑥))
6		eleq2 2878	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∪ ran 𝑦 ∈ 𝑢 ↔ ∪ ran 𝑦 ∈ 𝑈))
7	5, 6	raleqbidv 3354	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢 ↔ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))
8	2, 4, 7	3anbi123d 1433	. . . 4 ⊢ (𝑢 = 𝑈 → ((𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢) ↔ (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈)))
9	8	raleqbi1dv 3356	. . 3 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢) ↔ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈)))
10	1, 9	anbi12d 633	. 2 ⊢ (𝑢 = 𝑈 → ((Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢)) ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))))
11		df-gru 10202	. 2 ⊢ Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢))}
12	10, 11	elab2g 3616	1 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  𝒫 cpw 4497  {cpr 4527  ∪ cuni 4800  Tr wtr 5136  ran crn 5520  (class class class)co 7135   ↑_m cmap 8389  Univcgru 10201

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

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 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-tr 5137  df-iota 6283  df-fv 6332  df-ov 7138  df-gru 10202

This theorem is referenced by:  grutr  10204  grupw  10206  grupr  10208  gruurn  10209  intgru  10225  ingru  10226  grutsk1  10232  mnugrud  40992