MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gruiun Structured version   Visualization version   GIF version

Theorem gruiun 9907
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 2797 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21fnmpt 6229 . . . . . 6 (∀𝑥𝐴 𝐵𝑈 → (𝑥𝐴𝐵) Fn 𝐴)
31rnmptss 6616 . . . . . 6 (∀𝑥𝐴 𝐵𝑈 → ran (𝑥𝐴𝐵) ⊆ 𝑈)
4 df-f 6103 . . . . . 6 ((𝑥𝐴𝐵):𝐴𝑈 ↔ ((𝑥𝐴𝐵) Fn 𝐴 ∧ ran (𝑥𝐴𝐵) ⊆ 𝑈))
52, 3, 4sylanbrc 579 . . . . 5 (∀𝑥𝐴 𝐵𝑈 → (𝑥𝐴𝐵):𝐴𝑈)
6 gruurn 9906 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (𝑥𝐴𝐵):𝐴𝑈) → ran (𝑥𝐴𝐵) ∈ 𝑈)
763expia 1151 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ((𝑥𝐴𝐵):𝐴𝑈 ran (𝑥𝐴𝐵) ∈ 𝑈))
85, 7syl5com 31 . . . 4 (∀𝑥𝐴 𝐵𝑈 → ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ran (𝑥𝐴𝐵) ∈ 𝑈))
9 dfiun3g 5580 . . . . 5 (∀𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
109eleq1d 2861 . . . 4 (∀𝑥𝐴 𝐵𝑈 → ( 𝑥𝐴 𝐵𝑈 ran (𝑥𝐴𝐵) ∈ 𝑈))
118, 10sylibrd 251 . . 3 (∀𝑥𝐴 𝐵𝑈 → ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝑥𝐴 𝐵𝑈))
1211com12 32 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (∀𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵𝑈))
13123impia 1146 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108  wcel 2157  wral 3087  wss 3767   cuni 4626   ciun 4708  cmpt 4920  ran crn 5311   Fn wfn 6094  wf 6095  Univcgru 9898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-map 8095  df-gru 9899
This theorem is referenced by:  gruuni  9908  gruun  9914  gruixp  9917  grur1a  9927
  Copyright terms: Public domain W3C validator