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Theorem grurn 10839
Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10837 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 1135 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝑈 ∈ Univ)
2 gruurn 10836 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
3 grupw 10833 . . 3 ((𝑈 ∈ Univ ∧ ran 𝐹𝑈) → 𝒫 ran 𝐹𝑈)
41, 2, 3syl2anc 584 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝒫 ran 𝐹𝑈)
5 pwuni 4950 . . 3 ran 𝐹 ⊆ 𝒫 ran 𝐹
65a1i 11 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹 ⊆ 𝒫 ran 𝐹)
7 gruss 10834 . 2 ((𝑈 ∈ Univ ∧ 𝒫 ran 𝐹𝑈 ∧ ran 𝐹 ⊆ 𝒫 ran 𝐹) → ran 𝐹𝑈)
81, 4, 6, 7syl3anc 1370 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2106  wss 3963  𝒫 cpw 4605   cuni 4912  ran crn 5690  wf 6559  Univcgru 10828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-gru 10829
This theorem is referenced by:  gruima  10840  gruf  10849  gruen  10850
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