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Theorem grurn 8678
Description: A Grothendieck's universe contains the range of any function which takes values in the universe (see gruiun 8676 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
grurn  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  ran  F  e.  U )

Proof of Theorem grurn
StepHypRef Expression
1 simp1 958 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  U  e.  Univ )
2 gruurn 8675 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  U. ran  F  e.  U )
3 grupw 8672 . . 3  |-  ( ( U  e.  Univ  /\  U. ran  F  e.  U )  ->  ~P U. ran  F  e.  U )
41, 2, 3syl2anc 644 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  ~P U.
ran  F  e.  U
)
5 pwuni 4397 . . 3  |-  ran  F  C_ 
~P U. ran  F
65a1i 11 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  ran  F 
C_  ~P U. ran  F
)
7 gruss 8673 . 2  |-  ( ( U  e.  Univ  /\  ~P U.
ran  F  e.  U  /\  ran  F  C_  ~P U.
ran  F )  ->  ran  F  e.  U )
81, 4, 6, 7syl3anc 1185 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  ran  F  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    e. wcel 1726    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   ran crn 4881   -->wf 5452   Univcgru 8667
This theorem is referenced by:  gruima  8679  gruf  8688  gruen  8689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-gru 8668
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