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Theorem gruiun 9659
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 2651 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21fnmpt 6058 . . . . . 6 (∀𝑥𝐴 𝐵𝑈 → (𝑥𝐴𝐵) Fn 𝐴)
31rnmptss 6432 . . . . . 6 (∀𝑥𝐴 𝐵𝑈 → ran (𝑥𝐴𝐵) ⊆ 𝑈)
4 df-f 5930 . . . . . 6 ((𝑥𝐴𝐵):𝐴𝑈 ↔ ((𝑥𝐴𝐵) Fn 𝐴 ∧ ran (𝑥𝐴𝐵) ⊆ 𝑈))
52, 3, 4sylanbrc 699 . . . . 5 (∀𝑥𝐴 𝐵𝑈 → (𝑥𝐴𝐵):𝐴𝑈)
6 gruurn 9658 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (𝑥𝐴𝐵):𝐴𝑈) → ran (𝑥𝐴𝐵) ∈ 𝑈)
763expia 1286 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ((𝑥𝐴𝐵):𝐴𝑈 ran (𝑥𝐴𝐵) ∈ 𝑈))
85, 7syl5com 31 . . . 4 (∀𝑥𝐴 𝐵𝑈 → ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ran (𝑥𝐴𝐵) ∈ 𝑈))
9 dfiun3g 5410 . . . . 5 (∀𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
109eleq1d 2715 . . . 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 1280 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wcel 2030  wral 2941  wss 3607   cuni 4468   ciun 4552  cmpt 4762  ran crn 5144   Fn wfn 5921  wf 5922  Univcgru 9650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-gru 9651
This theorem is referenced by:  gruuni  9660  gruun  9666  gruixp  9669  grur1a  9679
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