Theorem gruiun 10486

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 2738	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	fnmpt 6557	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
3	1	rnmptss 6978	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑈)
4		df-f 6422	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑈))
5	2, 3, 4	sylanbrc 582	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈)
6		gruurn 10485	. . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈)
7	6	3expia 1119	. . . . 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 5862	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
10	9	eleq1d 2823	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 ↔ ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈))
11	8, 10	sylibrd 258	. . 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 1115	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  ∪ cuni 4836  ∪ ciun 4921   ↦ cmpt 5153  ran crn 5581   Fn wfn 6413  ⟶wf 6414  Univcgru 10477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-gru 10478

This theorem is referenced by:  gruuni  10487  gruun  10493  gruixp  10496  grur1a  10506  grur1cld  41739  grucollcld  41767