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Theorem chfnrn 5991
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 5914 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  y ) )
21biimpd 207 . . . 4  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F `  x )  =  y ) )
3 eleq1 2539 . . . . . . 7  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  x  <->  y  e.  x ) )
43biimpcd 224 . . . . . 6  |-  ( ( F `  x )  e.  x  ->  (
( F `  x
)  =  y  -> 
y  e.  x ) )
54ralimi 2857 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  x  ->  A. x  e.  A  ( ( F `  x )  =  y  ->  y  e.  x ) )
6 rexim 2929 . . . . 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 16 . . . 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 657 . . 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 4249 . . 3  |-  ( y  e.  U. A  <->  E. x  e.  A  y  e.  x )
108, 9syl6ibr 227 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  x )  -> 
( y  e.  ran  F  ->  y  e.  U. A ) )
1110ssrdv 3510 1  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  x )  ->  ran  F  C_  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   U.cuni 4245   ran crn 5000    Fn wfn 5582   ` cfv 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5550  df-fun 5589  df-fn 5590  df-fv 5595
This theorem is referenced by:  stoweidlem59  31375
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