Theorem intgru 10737

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 484	. . 3 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
2		intex 5291	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
3	1, 2	sylib 218	. 2 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ V)
4		dfss3 3924	. . . . 5 ⊢ (𝐴 ⊆ Univ ↔ ∀𝑢 ∈ 𝐴 𝑢 ∈ Univ)
5		grutr 10716	. . . . . 6 ⊢ (𝑢 ∈ Univ → Tr 𝑢)
6	5	ralimi 3075	. . . . 5 ⊢ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ → ∀𝑢 ∈ 𝐴 Tr 𝑢)
7	4, 6	sylbi 217	. . . 4 ⊢ (𝐴 ⊆ Univ → ∀𝑢 ∈ 𝐴 Tr 𝑢)
8		trint 5224	. . . 4 ⊢ (∀𝑢 ∈ 𝐴 Tr 𝑢 → Tr ∩ 𝐴)
9	7, 8	syl 17	. . 3 ⊢ (𝐴 ⊆ Univ → Tr ∩ 𝐴)
10	9	adantr 480	. 2 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → Tr ∩ 𝐴)
11		grupw 10718	. . . . . . . . . 10 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → 𝒫 𝑥 ∈ 𝑢)
12	11	ex 412	. . . . . . . . 9 ⊢ (𝑢 ∈ Univ → (𝑥 ∈ 𝑢 → 𝒫 𝑥 ∈ 𝑢))
13	12	ral2imi 3077	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ → (∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢 → ∀𝑢 ∈ 𝐴 𝒫 𝑥 ∈ 𝑢))
14		vex 3446	. . . . . . . . 9 ⊢ 𝑥 ∈ V
15	14	elint2 4911	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
16		vpwex 5324	. . . . . . . . 9 ⊢ 𝒫 𝑥 ∈ V
17	16	elint2 4911	. . . . . . . 8 ⊢ (𝒫 𝑥 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 𝒫 𝑥 ∈ 𝑢)
18	13, 15, 17	3imtr4g 296	. . . . . . 7 ⊢ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ → (𝑥 ∈ ∩ 𝐴 → 𝒫 𝑥 ∈ ∩ 𝐴))
19	18	imp 406	. . . . . 6 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → 𝒫 𝑥 ∈ ∩ 𝐴)
20	19	adantlr 716	. . . . 5 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → 𝒫 𝑥 ∈ ∩ 𝐴)
21		r19.26 3098	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝐴 (𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) ↔ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢))
22		grupr 10720	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) → {𝑥, 𝑦} ∈ 𝑢)
23	22	3expia 1122	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (𝑦 ∈ 𝑢 → {𝑥, 𝑦} ∈ 𝑢))
24	23	ral2imi 3077	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝐴 (𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦 ∈ 𝑢 → ∀𝑢 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑢))
25	21, 24	sylbir 235	. . . . . . . . 9 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦 ∈ 𝑢 → ∀𝑢 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑢))
26		vex 3446	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
27	26	elint2 4911	. . . . . . . . 9 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 𝑦 ∈ 𝑢)
28		prex 5384	. . . . . . . . . 10 ⊢ {𝑥, 𝑦} ∈ V
29	28	elint2 4911	. . . . . . . . 9 ⊢ ({𝑥, 𝑦} ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑢)
30	25, 27, 29	3imtr4g 296	. . . . . . . 8 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) → (𝑦 ∈ ∩ 𝐴 → {𝑥, 𝑦} ∈ ∩ 𝐴))
31	15, 30	sylan2b 595	. . . . . . 7 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦 ∈ ∩ 𝐴 → {𝑥, 𝑦} ∈ ∩ 𝐴))
32	31	ralrimiv 3129	. . . . . 6 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴)
33	32	adantlr 716	. . . . 5 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴)
34		elmapg 8788	. . . . . . . . . 10 ⊢ ((∩ 𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
35	34	elvd 3448	. . . . . . . . 9 ⊢ (∩ 𝐴 ∈ V → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
36	2, 35	sylbi 217	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
37	36	ad2antlr 728	. . . . . . 7 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
38		intss1 4920	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑢)
39		fss 6686	. . . . . . . . . . . 12 ⊢ ((𝑦:𝑥⟶∩ 𝐴 ∧ ∩ 𝐴 ⊆ 𝑢) → 𝑦:𝑥⟶𝑢)
40	38, 39	sylan2 594	. . . . . . . . . . 11 ⊢ ((𝑦:𝑥⟶∩ 𝐴 ∧ 𝑢 ∈ 𝐴) → 𝑦:𝑥⟶𝑢)
41	40	ralrimiva 3130	. . . . . . . . . 10 ⊢ (𝑦:𝑥⟶∩ 𝐴 → ∀𝑢 ∈ 𝐴 𝑦:𝑥⟶𝑢)
42		gruurn 10721	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢 ∧ 𝑦:𝑥⟶𝑢) → ∪ ran 𝑦 ∈ 𝑢)
43	42	3expia 1122	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (𝑦:𝑥⟶𝑢 → ∪ ran 𝑦 ∈ 𝑢))
44	43	ral2imi 3077	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ 𝐴 (𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦:𝑥⟶𝑢 → ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢))
45	21, 44	sylbir 235	. . . . . . . . . . 11 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦:𝑥⟶𝑢 → ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢))
46	15, 45	sylan2b 595	. . . . . . . . . 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 7862	. . . . . . . . . . 11 ⊢ ran 𝑦 ∈ V
49	48	uniex 7696	. . . . . . . . . 10 ⊢ ∪ ran 𝑦 ∈ V
50	49	elint2 4911	. . . . . . . . 9 ⊢ (∪ ran 𝑦 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢)
51	47, 50	imbitrrdi 252	. . . . . . . 8 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦:𝑥⟶∩ 𝐴 → ∪ ran 𝑦 ∈ ∩ 𝐴))
52	51	adantlr 716	. . . . . . 7 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦:𝑥⟶∩ 𝐴 → ∪ ran 𝑦 ∈ ∩ 𝐴))
53	37, 52	sylbid 240	. . . . . 6 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) → ∪ ran 𝑦 ∈ ∩ 𝐴))
54	53	ralrimiv 3129	. . . . 5 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴)
55	20, 33, 54	3jca 1129	. . . 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 3130	. . 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 582	. 2 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 ∈ ∩ 𝐴(𝒫 𝑥 ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴))
58		elgrug 10715	. . 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 477	. 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 837	1 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Univ)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {cpr 4584  ∪ cuni 4865  ∩ cint 4904  Tr wtr 5207  ran crn 5633  ⟶wf 6496  (class class class)co 7368   ↑_m cmap 8775  Univcgru 10713

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iin 4951  df-br 5101  df-opab 5163  df-tr 5208  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-gru 10714

This theorem is referenced by: (None)