Theorem elgrug 10786

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 5266	. . 3 ⊢ (𝑢 = 𝑈 → (Tr 𝑢 ↔ Tr 𝑈))
2		eleq2 2816	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝒫 𝑥 ∈ 𝑢 ↔ 𝒫 𝑥 ∈ 𝑈))
3		eleq2 2816	. . . . . 6 ⊢ (𝑢 = 𝑈 → ({𝑥, 𝑦} ∈ 𝑢 ↔ {𝑥, 𝑦} ∈ 𝑈))
4	3	raleqbi1dv 3327	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ↔ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
5		oveq1 7411	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 ↑m 𝑥) = (𝑈 ↑m 𝑥))
6		eleq2 2816	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∪ ran 𝑦 ∈ 𝑢 ↔ ∪ ran 𝑦 ∈ 𝑈))
7	5, 6	raleqbidv 3336	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢 ↔ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))
8	2, 4, 7	3anbi123d 1432	. . . 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 3327	. . 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 630	. 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 10785	. 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 3665	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 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3055  𝒫 cpw 4597  {cpr 4625  ∪ cuni 4902  Tr wtr 5258  ran crn 5670  (class class class)co 7404   ↑_m cmap 8819  Univcgru 10784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-tr 5259  df-iota 6488  df-fv 6544  df-ov 7407  df-gru 10785

This theorem is referenced by:  grutr  10787  grupw  10789  grupr  10791  gruurn  10792  intgru  10808  ingru  10809  grutsk1  10815  mnugrud  43600