MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  chfnrn Unicode version

Theorem chfnrn 5719
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 5653 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  y ) )
21biimpd 198 . . . 4  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F `  x )  =  y ) )
3 eleq1 2418 . . . . . . 7  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  x  <->  y  e.  x ) )
43biimpcd 215 . . . . . 6  |-  ( ( F `  x )  e.  x  ->  (
( F `  x
)  =  y  -> 
y  e.  x ) )
54ralimi 2694 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  x  ->  A. x  e.  A  ( ( F `  x )  =  y  ->  y  e.  x ) )
6 rexim 2723 . . . . 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 15 . . . 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 638 . . 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 3912 . . 3  |-  ( y  e.  U. A  <->  E. x  e.  A  y  e.  x )
108, 9syl6ibr 218 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  x )  -> 
( y  e.  ran  F  ->  y  e.  U. A ) )
1110ssrdv 3261 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 358    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620    C_ wss 3228   U.cuni 3908   ran crn 4772    Fn wfn 5332   ` cfv 5337
This theorem is referenced by:  stoweidlem59  27131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345
  Copyright terms: Public domain W3C validator