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Theorem gruiun 10794
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 2733 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21fnmpt 6691 . . . . . 6 (∀𝑥𝐴 𝐵𝑈 → (𝑥𝐴𝐵) Fn 𝐴)
31rnmptss 7122 . . . . . 6 (∀𝑥𝐴 𝐵𝑈 → ran (𝑥𝐴𝐵) ⊆ 𝑈)
4 df-f 6548 . . . . . 6 ((𝑥𝐴𝐵):𝐴𝑈 ↔ ((𝑥𝐴𝐵) Fn 𝐴 ∧ ran (𝑥𝐴𝐵) ⊆ 𝑈))
52, 3, 4sylanbrc 584 . . . . 5 (∀𝑥𝐴 𝐵𝑈 → (𝑥𝐴𝐵):𝐴𝑈)
6 gruurn 10793 . . . . . 6 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (𝑥𝐴𝐵):𝐴𝑈) → ran (𝑥𝐴𝐵) ∈ 𝑈)
763expia 1122 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ((𝑥𝐴𝐵):𝐴𝑈 ran (𝑥𝐴𝐵) ∈ 𝑈))
85, 7syl5com 31 . . . 4 (∀𝑥𝐴 𝐵𝑈 → ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ran (𝑥𝐴𝐵) ∈ 𝑈))
9 dfiun3g 5964 . . . . 5 (∀𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
109eleq1d 2819 . . . 4 (∀𝑥𝐴 𝐵𝑈 → ( 𝑥𝐴 𝐵𝑈 ran (𝑥𝐴𝐵) ∈ 𝑈))
118, 10sylibrd 259 . . 3 (∀𝑥𝐴 𝐵𝑈 → ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝑥𝐴 𝐵𝑈))
1211com12 32 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (∀𝑥𝐴 𝐵𝑈 𝑥𝐴 𝐵𝑈))
13123impia 1118 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  wral 3062  wss 3949   cuni 4909   ciun 4998  cmpt 5232  ran crn 5678   Fn wfn 6539  wf 6540  Univcgru 10785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-gru 10786
This theorem is referenced by:  gruuni  10795  gruun  10801  gruixp  10804  grur1a  10814  grur1cld  42991  grucollcld  43019
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