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Theorem gruiun 10209
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 2818 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21fnmpt 6481 . . . . . 6 (∀𝑥𝐴 𝐵𝑈 → (𝑥𝐴𝐵) Fn 𝐴)
31rnmptss 6878 . . . . . 6 (∀𝑥𝐴 𝐵𝑈 → ran (𝑥𝐴𝐵) ⊆ 𝑈)
4 df-f 6352 . . . . . 6 ((𝑥𝐴𝐵):𝐴𝑈 ↔ ((𝑥𝐴𝐵) Fn 𝐴 ∧ ran (𝑥𝐴𝐵) ⊆ 𝑈))
52, 3, 4sylanbrc 583 . . . . 5 (∀𝑥𝐴 𝐵𝑈 → (𝑥𝐴𝐵):𝐴𝑈)
6 gruurn 10208 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (𝑥𝐴𝐵):𝐴𝑈) → ran (𝑥𝐴𝐵) ∈ 𝑈)
763expia 1113 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ((𝑥𝐴𝐵):𝐴𝑈 ran (𝑥𝐴𝐵) ∈ 𝑈))
85, 7syl5com 31 . . . 4 (∀𝑥𝐴 𝐵𝑈 → ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ran (𝑥𝐴𝐵) ∈ 𝑈))
9 dfiun3g 5828 . . . . 5 (∀𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
109eleq1d 2894 . . . 4 (∀𝑥𝐴 𝐵𝑈 → ( 𝑥𝐴 𝐵𝑈 ran (𝑥𝐴𝐵) ∈ 𝑈))
118, 10sylibrd 260 . . 3 (∀𝑥𝐴 𝐵𝑈 → ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝑥𝐴 𝐵𝑈))
1211com12 32 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (∀𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵𝑈))
13123impia 1109 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079  wcel 2105  wral 3135  wss 3933   cuni 4830   ciun 4910  cmpt 5137  ran crn 5549   Fn wfn 6343  wf 6344  Univcgru 10200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-gru 10201
This theorem is referenced by:  gruuni  10210  gruun  10216  gruixp  10219  grur1a  10229  grur1cld  40445  grucollcld  40473
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