Theorem gruiun 10839

Description: If 𝐵(𝑥) is a family of elements of 𝑈 and the index set 𝐴 is an element of 𝑈, then the indexed union ∪ 𝑥 ∈ 𝐴𝐵 is also an element of 𝑈, where 𝑈 is a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
gruiun	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)

Distinct variable groups:   𝑥,𝑈   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem gruiun

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	fnmpt 6708	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
3	1	rnmptss 7143	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑈)
4		df-f 6565	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑈))
5	2, 3, 4	sylanbrc 583	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈)
6		gruurn 10838	. . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈)
7	6	3expia 1122	. . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈 → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈))
8	5, 7	syl5com 31	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈))
9		dfiun3g 5978	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
10	9	eleq1d 2826	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 ↔ ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈))
11	8, 10	sylibrd 259	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈))
12	11	com12 32	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈))
13	12	3impia 1118	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  ∪ cuni 4907  ∪ ciun 4991   ↦ cmpt 5225  ran crn 5686   Fn wfn 6556  ⟶wf 6557  Univcgru 10830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-gru 10831

This theorem is referenced by:  gruuni  10840  gruun  10846  gruixp  10849  grur1a  10859  grur1cld  44251  grucollcld  44279