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

Theorem chfnrn 5804
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 5737 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  y ) )
21biimpd 199 . . . 4  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F `  x )  =  y ) )
3 eleq1 2468 . . . . . . 7  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  x  <->  y  e.  x ) )
43biimpcd 216 . . . . . 6  |-  ( ( F `  x )  e.  x  ->  (
( F `  x
)  =  y  -> 
y  e.  x ) )
54ralimi 2745 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  x  ->  A. x  e.  A  ( ( F `  x )  =  y  ->  y  e.  x ) )
6 rexim 2774 . . . . 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 639 . . 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 3983 . . 3  |-  ( y  e.  U. A  <->  E. x  e.  A  y  e.  x )
108, 9syl6ibr 219 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  x )  -> 
( y  e.  ran  F  ->  y  e.  U. A ) )
1110ssrdv 3318 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 359    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671    C_ wss 3284   U.cuni 3979   ran crn 4842    Fn wfn 5412   ` cfv 5417
This theorem is referenced by:  stoweidlem59  27679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-fv 5425
  Copyright terms: Public domain W3C validator