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Theorem grurn 10802
Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10800 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 10799 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
3 grupw 10796 . . 3 ((𝑈 ∈ Univ ∧ ran 𝐹𝑈) → 𝒫 ran 𝐹𝑈)
41, 2, 3syl2anc 583 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝒫 ran 𝐹𝑈)
5 pwuni 4949 . . 3 ran 𝐹 ⊆ 𝒫 ran 𝐹
65a1i 11 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹 ⊆ 𝒫 ran 𝐹)
7 gruss 10797 . 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 2105  wss 3948  𝒫 cpw 4602   cuni 4908  ran crn 5677  wf 6539  Univcgru 10791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-gru 10792
This theorem is referenced by:  gruima  10803  gruf  10812  gruen  10813
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