Theorem intgru 10843

Description: The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
intgru	⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Univ)

Proof of Theorem intgru

Dummy variables 𝑥 𝑢 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 483	. . 3 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
2		intex 5341	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
3	1, 2	sylib 217	. 2 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ V)
4		dfss3 3968	. . . . 5 ⊢ (𝐴 ⊆ Univ ↔ ∀𝑢 ∈ 𝐴 𝑢 ∈ Univ)
5		grutr 10822	. . . . . 6 ⊢ (𝑢 ∈ Univ → Tr 𝑢)
6	5	ralimi 3079	. . . . 5 ⊢ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ → ∀𝑢 ∈ 𝐴 Tr 𝑢)
7	4, 6	sylbi 216	. . . 4 ⊢ (𝐴 ⊆ Univ → ∀𝑢 ∈ 𝐴 Tr 𝑢)
8		trint 5285	. . . 4 ⊢ (∀𝑢 ∈ 𝐴 Tr 𝑢 → Tr ∩ 𝐴)
9	7, 8	syl 17	. . 3 ⊢ (𝐴 ⊆ Univ → Tr ∩ 𝐴)
10	9	adantr 479	. 2 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → Tr ∩ 𝐴)
11		grupw 10824	. . . . . . . . . 10 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → 𝒫 𝑥 ∈ 𝑢)
12	11	ex 411	. . . . . . . . 9 ⊢ (𝑢 ∈ Univ → (𝑥 ∈ 𝑢 → 𝒫 𝑥 ∈ 𝑢))
13	12	ral2imi 3081	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ → (∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢 → ∀𝑢 ∈ 𝐴 𝒫 𝑥 ∈ 𝑢))
14		vex 3475	. . . . . . . . 9 ⊢ 𝑥 ∈ V
15	14	elint2 4958	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
16		vpwex 5379	. . . . . . . . 9 ⊢ 𝒫 𝑥 ∈ V
17	16	elint2 4958	. . . . . . . 8 ⊢ (𝒫 𝑥 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 𝒫 𝑥 ∈ 𝑢)
18	13, 15, 17	3imtr4g 295	. . . . . . 7 ⊢ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ → (𝑥 ∈ ∩ 𝐴 → 𝒫 𝑥 ∈ ∩ 𝐴))
19	18	imp 405	. . . . . 6 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → 𝒫 𝑥 ∈ ∩ 𝐴)
20	19	adantlr 713	. . . . 5 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → 𝒫 𝑥 ∈ ∩ 𝐴)
21		r19.26 3107	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝐴 (𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) ↔ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢))
22		grupr 10826	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) → {𝑥, 𝑦} ∈ 𝑢)
23	22	3expia 1118	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (𝑦 ∈ 𝑢 → {𝑥, 𝑦} ∈ 𝑢))
24	23	ral2imi 3081	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝐴 (𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦 ∈ 𝑢 → ∀𝑢 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑢))
25	21, 24	sylbir 234	. . . . . . . . 9 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦 ∈ 𝑢 → ∀𝑢 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑢))
26		vex 3475	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
27	26	elint2 4958	. . . . . . . . 9 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 𝑦 ∈ 𝑢)
28		prex 5436	. . . . . . . . . 10 ⊢ {𝑥, 𝑦} ∈ V
29	28	elint2 4958	. . . . . . . . 9 ⊢ ({𝑥, 𝑦} ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑢)
30	25, 27, 29	3imtr4g 295	. . . . . . . 8 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) → (𝑦 ∈ ∩ 𝐴 → {𝑥, 𝑦} ∈ ∩ 𝐴))
31	15, 30	sylan2b 592	. . . . . . 7 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦 ∈ ∩ 𝐴 → {𝑥, 𝑦} ∈ ∩ 𝐴))
32	31	ralrimiv 3141	. . . . . 6 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴)
33	32	adantlr 713	. . . . 5 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴)
34		elmapg 8862	. . . . . . . . . 10 ⊢ ((∩ 𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
35	34	elvd 3478	. . . . . . . . 9 ⊢ (∩ 𝐴 ∈ V → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
36	2, 35	sylbi 216	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
37	36	ad2antlr 725	. . . . . . 7 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
38		intss1 4968	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑢)
39		fss 6742	. . . . . . . . . . . 12 ⊢ ((𝑦:𝑥⟶∩ 𝐴 ∧ ∩ 𝐴 ⊆ 𝑢) → 𝑦:𝑥⟶𝑢)
40	38, 39	sylan2 591	. . . . . . . . . . 11 ⊢ ((𝑦:𝑥⟶∩ 𝐴 ∧ 𝑢 ∈ 𝐴) → 𝑦:𝑥⟶𝑢)
41	40	ralrimiva 3142	. . . . . . . . . 10 ⊢ (𝑦:𝑥⟶∩ 𝐴 → ∀𝑢 ∈ 𝐴 𝑦:𝑥⟶𝑢)
42		gruurn 10827	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢 ∧ 𝑦:𝑥⟶𝑢) → ∪ ran 𝑦 ∈ 𝑢)
43	42	3expia 1118	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (𝑦:𝑥⟶𝑢 → ∪ ran 𝑦 ∈ 𝑢))
44	43	ral2imi 3081	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ 𝐴 (𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦:𝑥⟶𝑢 → ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢))
45	21, 44	sylbir 234	. . . . . . . . . . 11 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦:𝑥⟶𝑢 → ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢))
46	15, 45	sylan2b 592	. . . . . . . . . 10 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → (∀𝑢 ∈ 𝐴 𝑦:𝑥⟶𝑢 → ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢))
47	41, 46	syl5 34	. . . . . . . . 9 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦:𝑥⟶∩ 𝐴 → ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢))
48	26	rnex 7922	. . . . . . . . . . 11 ⊢ ran 𝑦 ∈ V
49	48	uniex 7750	. . . . . . . . . 10 ⊢ ∪ ran 𝑦 ∈ V
50	49	elint2 4958	. . . . . . . . 9 ⊢ (∪ ran 𝑦 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢)
51	47, 50	imbitrrdi 251	. . . . . . . 8 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦:𝑥⟶∩ 𝐴 → ∪ ran 𝑦 ∈ ∩ 𝐴))
52	51	adantlr 713	. . . . . . 7 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦:𝑥⟶∩ 𝐴 → ∪ ran 𝑦 ∈ ∩ 𝐴))
53	37, 52	sylbid 239	. . . . . 6 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) → ∪ ran 𝑦 ∈ ∩ 𝐴))
54	53	ralrimiv 3141	. . . . 5 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴)
55	20, 33, 54	3jca 1125	. . . 4 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → (𝒫 𝑥 ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴))
56	55	ralrimiva 3142	. . 3 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 ∈ ∩ 𝐴(𝒫 𝑥 ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴))
57	4, 56	sylanb 579	. 2 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 ∈ ∩ 𝐴(𝒫 𝑥 ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴))
58		elgrug 10821	. . 3 ⊢ (∩ 𝐴 ∈ V → (∩ 𝐴 ∈ Univ ↔ (Tr ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴(𝒫 𝑥 ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴))))
59	58	biimpar 476	. 2 ⊢ ((∩ 𝐴 ∈ V ∧ (Tr ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴(𝒫 𝑥 ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴))) → ∩ 𝐴 ∈ Univ)
60	3, 10, 57, 59	syl12anc 835	1 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Univ)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2098   ≠ wne 2936  ∀wral 3057  Vcvv 3471   ⊆ wss 3947  ∅c0 4324  𝒫 cpw 4604  {cpr 4632  ∪ cuni 4910  ∩ cint 4951  Tr wtr 5267  ran crn 5681  ⟶wf 6547  (class class class)co 7424   ↑_m cmap 8849  Univcgru 10819

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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iin 5001  df-br 5151  df-opab 5213  df-tr 5268  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8851  df-gru 10820

This theorem is referenced by: (None)