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Theorem grurn 8439
Description: A Grothendieck's universe contains the range of any function which takes values in the universe (see gruiun 8437 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 955 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  U  e.  Univ )
2 gruurn 8436 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  U. ran  F  e.  U )
3 grupw 8433 . . 3  |-  ( ( U  e.  Univ  /\  U. ran  F  e.  U )  ->  ~P U. ran  F  e.  U )
41, 2, 3syl2anc 642 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  ~P U.
ran  F  e.  U
)
5 pwuni 4222 . . 3  |-  ran  F  C_ 
~P U. ran  F
65a1i 10 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  ran  F 
C_  ~P U. ran  F
)
7 gruss 8434 . 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 1182 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 934    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   ran crn 4706   -->wf 5267   Univcgru 8428
This theorem is referenced by:  gruima  8440  gruf  8449  gruen  8450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-gru 8429
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