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Theorem grurn 10200
Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10198 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 1133 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝑈 ∈ Univ)
2 gruurn 10197 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
3 grupw 10194 . . 3 ((𝑈 ∈ Univ ∧ ran 𝐹𝑈) → 𝒫 ran 𝐹𝑈)
41, 2, 3syl2anc 587 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝒫 ran 𝐹𝑈)
5 pwuni 4848 . . 3 ran 𝐹 ⊆ 𝒫 ran 𝐹
65a1i 11 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹 ⊆ 𝒫 ran 𝐹)
7 gruss 10195 . 2 ((𝑈 ∈ Univ ∧ 𝒫 ran 𝐹𝑈 ∧ ran 𝐹 ⊆ 𝒫 ran 𝐹) → ran 𝐹𝑈)
81, 4, 6, 7syl3anc 1368 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2115  wss 3910  𝒫 cpw 4512   cuni 4811  ran crn 5529  wf 6324  Univcgru 10189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-tr 5146  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-gru 10190
This theorem is referenced by:  gruima  10201  gruf  10210  gruen  10211
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