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Theorem chfnrn 5410
Description: The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)
Assertion
Ref Expression
chfnrn  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  x )  ->  ran  F  C_  U. A )
Distinct variable groups:    x, A    x, F

Proof of Theorem chfnrn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fvelrnb 5352 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  y ) )
21biimpd 142 . . . 4  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F `  x )  =  y ) )
3 eleq1 2150 . . . . . . 7  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  x  <->  y  e.  x ) )
43biimpcd 157 . . . . . 6  |-  ( ( F `  x )  e.  x  ->  (
( F `  x
)  =  y  -> 
y  e.  x ) )
54ralimi 2438 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  x  ->  A. x  e.  A  ( ( F `  x )  =  y  ->  y  e.  x ) )
6 rexim 2467 . . . . 5  |-  ( A. x  e.  A  (
( F `  x
)  =  y  -> 
y  e.  x )  ->  ( E. x  e.  A  ( F `  x )  =  y  ->  E. x  e.  A  y  e.  x )
)
75, 6syl 14 . . . 4  |-  ( A. x  e.  A  ( F `  x )  e.  x  ->  ( E. x  e.  A  ( F `  x )  =  y  ->  E. x  e.  A  y  e.  x ) )
82, 7sylan9 401 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  x )  -> 
( y  e.  ran  F  ->  E. x  e.  A  y  e.  x )
)
9 eluni2 3657 . . 3  |-  ( y  e.  U. A  <->  E. x  e.  A  y  e.  x )
108, 9syl6ibr 160 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  x )  -> 
( y  e.  ran  F  ->  y  e.  U. A ) )
1110ssrdv 3031 1  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  x )  ->  ran  F  C_  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360    C_ wss 2999   U.cuni 3653   ran crn 4439    Fn wfn 5010   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by: (None)
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