Theorem gruiun 10787

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 2770	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	fnmpt 6679	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
3	1	rnmptss 7122	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑈)
4		df-f 6544	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑈))
5	2, 3, 4	sylanbrc 594	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈)
6		gruurn 10786	. . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈)
7	6	3expia 1137	. . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈 → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈))
8	5, 7	syl5com 32	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈))
9		dfiun3g 5962	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
10	9	eleq1d 2855	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 ↔ ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈))
11	8, 10	sylibrd 262	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈))
12	11	com12 33	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈))
13	12	3impia 1133	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   ∈ wcel 2150  ∀wral 3086   ⊆ wss 3913  ∪ cuni 4877  ∪ ciun 4961   ↦ cmpt 5197  ran crn 5666   Fn wfn 6535  ⟶wf 6536  Univcgru 10778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8829  df-gru 10779

This theorem is referenced by:  gruuni  10788  gruun  10794  gruixp  10797  grur1a  10807  grur1cld  44908  grucollcld  44922