Theorem gruiun 10724

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 6642	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
3	1	rnmptss 7079	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑈)
4		df-f 6506	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑈))
5	2, 3, 4	sylanbrc 584	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈)
6		gruurn 10723	. . . . . 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 5927	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
10	9	eleq1d 2822	. . . 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 2114  ∀wral 3052   ⊆ wss 3903  ∪ cuni 4865  ∪ ciun 4948   ↦ cmpt 5181  ran crn 5635   Fn wfn 6497  ⟶wf 6498  Univcgru 10715

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 5245  ax-pow 5314  ax-pr 5381  ax-un 7692

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-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-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-map 8779  df-gru 10716

This theorem is referenced by:  gruuni  10725  gruun  10731  gruixp  10734  grur1a  10744  grur1cld  44617  grucollcld  44645