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Theorem chfnrn 6015
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 5934 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  y ) )
21biimpd 212 . . . 4  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F `  x )  =  y ) )
3 eleq1 2527 . . . . . . 7  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  x  <->  y  e.  x ) )
43biimpcd 232 . . . . . 6  |-  ( ( F `  x )  e.  x  ->  (
( F `  x
)  =  y  -> 
y  e.  x ) )
54ralimi 2792 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  x  ->  A. x  e.  A  ( ( F `  x )  =  y  ->  y  e.  x ) )
6 rexim 2863 . . . . 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 17 . . . 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 667 . . 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 4215 . . 3  |-  ( y  e.  U. A  <->  E. x  e.  A  y  e.  x )
108, 9syl6ibr 235 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  x )  -> 
( y  e.  ran  F  ->  y  e.  U. A ) )
1110ssrdv 3449 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 375    = wceq 1454    e. wcel 1897   A.wral 2748   E.wrex 2749    C_ wss 3415   U.cuni 4211   ran crn 4853    Fn wfn 5595   ` cfv 5600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5564  df-fun 5602  df-fn 5603  df-fv 5608
This theorem is referenced by:  stoweidlem59  37957
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