Theorem intgru 10735

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		intex 5279	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
2	1	bilani 505	. 2 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ V)
3		dfss3 3911	. . . . 5 ⊢ (𝐴 ⊆ Univ ↔ ∀𝑢 ∈ 𝐴 𝑢 ∈ Univ)
4		grutr 10714	. . . . . 6 ⊢ (𝑢 ∈ Univ → Tr 𝑢)
5	4	ralimi 3077	. . . . 5 ⊢ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ → ∀𝑢 ∈ 𝐴 Tr 𝑢)
6	3, 5	sylbi 218	. . . 4 ⊢ (𝐴 ⊆ Univ → ∀𝑢 ∈ 𝐴 Tr 𝑢)
7		trint 5204	. . . 4 ⊢ (∀𝑢 ∈ 𝐴 Tr 𝑢 → Tr ∩ 𝐴)
8	6, 7	syl 17	. . 3 ⊢ (𝐴 ⊆ Univ → Tr ∩ 𝐴)
9	8	adantr 481	. 2 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → Tr ∩ 𝐴)
10		grupw 10716	. . . . . . . . . 10 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → 𝒫 𝑥 ∈ 𝑢)
11	10	ex 413	. . . . . . . . 9 ⊢ (𝑢 ∈ Univ → (𝑥 ∈ 𝑢 → 𝒫 𝑥 ∈ 𝑢))
12	11	ral2imi 3079	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ → (∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢 → ∀𝑢 ∈ 𝐴 𝒫 𝑥 ∈ 𝑢))
13		vex 3436	. . . . . . . . 9 ⊢ 𝑥 ∈ V
14	13	elint2 4891	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
15		vpwex 5313	. . . . . . . . 9 ⊢ 𝒫 𝑥 ∈ V
16	15	elint2 4891	. . . . . . . 8 ⊢ (𝒫 𝑥 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 𝒫 𝑥 ∈ 𝑢)
17	12, 14, 16	3imtr4g 297	. . . . . . 7 ⊢ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ → (𝑥 ∈ ∩ 𝐴 → 𝒫 𝑥 ∈ ∩ 𝐴))
18	17	imp 407	. . . . . 6 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → 𝒫 𝑥 ∈ ∩ 𝐴)
19	18	adantlr 721	. . . . 5 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → 𝒫 𝑥 ∈ ∩ 𝐴)
20		r19.26 3100	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝐴 (𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) ↔ (∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢))
21		grupr 10718	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢) → {𝑥, 𝑦} ∈ 𝑢)
22	21	3expia 1127	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (𝑦 ∈ 𝑢 → {𝑥, 𝑦} ∈ 𝑢))
23	22	ral2imi 3079	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝐴 (𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦 ∈ 𝑢 → ∀𝑢 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑢))
24	20, 23	sylbir 236	. . . . . . . . 9 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦 ∈ 𝑢 → ∀𝑢 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑢))
25		vex 3436	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
26	25	elint2 4891	. . . . . . . . 9 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 𝑦 ∈ 𝑢)
27		prex 5374	. . . . . . . . . 10 ⊢ {𝑥, 𝑦} ∈ V
28	27	elint2 4891	. . . . . . . . 9 ⊢ ({𝑥, 𝑦} ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑢)
29	24, 26, 28	3imtr4g 297	. . . . . . . 8 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) → (𝑦 ∈ ∩ 𝐴 → {𝑥, 𝑦} ∈ ∩ 𝐴))
30	14, 29	sylan2b 600	. . . . . . 7 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦 ∈ ∩ 𝐴 → {𝑥, 𝑦} ∈ ∩ 𝐴))
31	30	ralrimiv 3131	. . . . . 6 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴)
32	31	adantlr 721	. . . . 5 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴)
33		elmapg 8783	. . . . . . . . . 10 ⊢ ((∩ 𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
34	33	elvd 3438	. . . . . . . . 9 ⊢ (∩ 𝐴 ∈ V → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
35	1, 34	sylbi 218	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
36	35	ad2antlr 733	. . . . . . 7 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) ↔ 𝑦:𝑥⟶∩ 𝐴))
37		intss1 4900	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑢)
38		fss 6678	. . . . . . . . . . . 12 ⊢ ((𝑦:𝑥⟶∩ 𝐴 ∧ ∩ 𝐴 ⊆ 𝑢) → 𝑦:𝑥⟶𝑢)
39	37, 38	sylan2 599	. . . . . . . . . . 11 ⊢ ((𝑦:𝑥⟶∩ 𝐴 ∧ 𝑢 ∈ 𝐴) → 𝑦:𝑥⟶𝑢)
40	39	ralrimiva 3132	. . . . . . . . . 10 ⊢ (𝑦:𝑥⟶∩ 𝐴 → ∀𝑢 ∈ 𝐴 𝑦:𝑥⟶𝑢)
41		gruurn 10719	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢 ∧ 𝑦:𝑥⟶𝑢) → ∪ ran 𝑦 ∈ 𝑢)
42	41	3expia 1127	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (𝑦:𝑥⟶𝑢 → ∪ ran 𝑦 ∈ 𝑢))
43	42	ral2imi 3079	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ 𝐴 (𝑢 ∈ Univ ∧ 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦:𝑥⟶𝑢 → ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢))
44	20, 43	sylbir 236	. . . . . . . . . . 11 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ ∀𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) → (∀𝑢 ∈ 𝐴 𝑦:𝑥⟶𝑢 → ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢))
45	14, 44	sylan2b 600	. . . . . . . . . 10 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → (∀𝑢 ∈ 𝐴 𝑦:𝑥⟶𝑢 → ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢))
46	40, 45	syl5 34	. . . . . . . . 9 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦:𝑥⟶∩ 𝐴 → ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢))
47	25	rnex 7857	. . . . . . . . . . 11 ⊢ ran 𝑦 ∈ V
48	47	uniex 7691	. . . . . . . . . 10 ⊢ ∪ ran 𝑦 ∈ V
49	48	elint2 4891	. . . . . . . . 9 ⊢ (∪ ran 𝑦 ∈ ∩ 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∪ ran 𝑦 ∈ 𝑢)
50	46, 49	imbitrrdi 253	. . . . . . . 8 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦:𝑥⟶∩ 𝐴 → ∪ ran 𝑦 ∈ ∩ 𝐴))
51	50	adantlr 721	. . . . . . 7 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦:𝑥⟶∩ 𝐴 → ∪ ran 𝑦 ∈ ∩ 𝐴))
52	36, 51	sylbid 241	. . . . . 6 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑦 ∈ (∩ 𝐴 ↑m 𝑥) → ∪ ran 𝑦 ∈ ∩ 𝐴))
53	52	ralrimiv 3131	. . . . 5 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴)
54	19, 32, 53	3jca 1134	. . . 4 ⊢ (((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ∩ 𝐴) → (𝒫 𝑥 ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴))
55	54	ralrimiva 3132	. . 3 ⊢ ((∀𝑢 ∈ 𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 ∈ ∩ 𝐴(𝒫 𝑥 ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴))
56	3, 55	sylanb 587	. 2 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 ∈ ∩ 𝐴(𝒫 𝑥 ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴))
57		elgrug 10713	. . 3 ⊢ (∩ 𝐴 ∈ V → (∩ 𝐴 ∈ Univ ↔ (Tr ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴(𝒫 𝑥 ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴))))
58	57	biimpar 478	. 2 ⊢ ((∩ 𝐴 ∈ V ∧ (Tr ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴(𝒫 𝑥 ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ ∩ 𝐴{𝑥, 𝑦} ∈ ∩ 𝐴 ∧ ∀𝑦 ∈ (∩ 𝐴 ↑m 𝑥)∪ ran 𝑦 ∈ ∩ 𝐴))) → ∩ 𝐴 ∈ Univ)
59	2, 9, 56, 58	syl12anc 842	1 ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Univ)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  Vcvv 3432   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536  {cpr 4564  ∪ cuni 4845  ∩ cint 4884  Tr wtr 5186  ran crn 5626  ⟶wf 6488  (class class class)co 7363   ↑_m cmap 8770  Univcgru 10711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iin 4931  df-br 5080  df-opab 5142  df-tr 5187  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-map 8772  df-gru 10712

This theorem is referenced by: (None)