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Theorem grurn 10841
Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10839 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
grurn ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)

Proof of Theorem grurn
StepHypRef Expression
1 simp1 1137 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝑈 ∈ Univ)
2 gruurn 10838 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
3 grupw 10835 . . 3 ((𝑈 ∈ Univ ∧ ran 𝐹𝑈) → 𝒫 ran 𝐹𝑈)
41, 2, 3syl2anc 584 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝒫 ran 𝐹𝑈)
5 pwuni 4945 . . 3 ran 𝐹 ⊆ 𝒫 ran 𝐹
65a1i 11 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹 ⊆ 𝒫 ran 𝐹)
7 gruss 10836 . 2 ((𝑈 ∈ Univ ∧ 𝒫 ran 𝐹𝑈 ∧ ran 𝐹 ⊆ 𝒫 ran 𝐹) → ran 𝐹𝑈)
81, 4, 6, 7syl3anc 1373 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2108  wss 3951  𝒫 cpw 4600   cuni 4907  ran crn 5686  wf 6557  Univcgru 10830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-gru 10831
This theorem is referenced by:  gruima  10842  gruf  10851  gruen  10852
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