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Theorem gruiun 9566
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 𝑥𝐴𝐵 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 ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
Distinct variable groups:   𝑥,𝑈   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem gruiun
StepHypRef Expression
1 eqid 2626 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21fnmpt 5979 . . . . . 6 (∀𝑥𝐴 𝐵𝑈 → (𝑥𝐴𝐵) Fn 𝐴)
31rnmptss 6348 . . . . . 6 (∀𝑥𝐴 𝐵𝑈 → ran (𝑥𝐴𝐵) ⊆ 𝑈)
4 df-f 5854 . . . . . 6 ((𝑥𝐴𝐵):𝐴𝑈 ↔ ((𝑥𝐴𝐵) Fn 𝐴 ∧ ran (𝑥𝐴𝐵) ⊆ 𝑈))
52, 3, 4sylanbrc 697 . . . . 5 (∀𝑥𝐴 𝐵𝑈 → (𝑥𝐴𝐵):𝐴𝑈)
6 gruurn 9565 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (𝑥𝐴𝐵):𝐴𝑈) → ran (𝑥𝐴𝐵) ∈ 𝑈)
763expia 1264 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ((𝑥𝐴𝐵):𝐴𝑈 ran (𝑥𝐴𝐵) ∈ 𝑈))
85, 7syl5com 31 . . . 4 (∀𝑥𝐴 𝐵𝑈 → ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ran (𝑥𝐴𝐵) ∈ 𝑈))
9 dfiun3g 5342 . . . . 5 (∀𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
109eleq1d 2688 . . . 4 (∀𝑥𝐴 𝐵𝑈 → ( 𝑥𝐴 𝐵𝑈 ran (𝑥𝐴𝐵) ∈ 𝑈))
118, 10sylibrd 249 . . 3 (∀𝑥𝐴 𝐵𝑈 → ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝑥𝐴 𝐵𝑈))
1211com12 32 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (∀𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵𝑈))
13123impia 1258 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1992  wral 2912  wss 3560   cuni 4407   ciun 4490  cmpt 4678  ran crn 5080   Fn wfn 5845  wf 5846  Univcgru 9557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-gru 9558
This theorem is referenced by:  gruuni  9567  gruun  9573  gruixp  9576  grur1a  9586
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