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Theorem fvelrn 5430
Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
fvelrn  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)

Proof of Theorem fvelrn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2150 . . . . 5  |-  ( x  =  A  ->  (
x  e.  dom  F  <->  A  e.  dom  F ) )
21anbi2d 452 . . . 4  |-  ( x  =  A  ->  (
( Fun  F  /\  x  e.  dom  F )  <-> 
( Fun  F  /\  A  e.  dom  F ) ) )
3 fveq2 5305 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
43eleq1d 2156 . . . 4  |-  ( x  =  A  ->  (
( F `  x
)  e.  ran  F  <->  ( F `  A )  e.  ran  F ) )
52, 4imbi12d 232 . . 3  |-  ( x  =  A  ->  (
( ( Fun  F  /\  x  e.  dom  F )  ->  ( F `  x )  e.  ran  F )  <->  ( ( Fun 
F  /\  A  e.  dom  F )  ->  ( F `  A )  e.  ran  F ) ) )
6 funfvop 5411 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. x ,  ( F `
 x ) >.  e.  F )
7 vex 2622 . . . . . 6  |-  x  e. 
_V
8 opeq1 3622 . . . . . . 7  |-  ( y  =  x  ->  <. y ,  ( F `  x ) >.  =  <. x ,  ( F `  x ) >. )
98eleq1d 2156 . . . . . 6  |-  ( y  =  x  ->  ( <. y ,  ( F `
 x ) >.  e.  F  <->  <. x ,  ( F `  x )
>.  e.  F ) )
107, 9spcev 2713 . . . . 5  |-  ( <.
x ,  ( F `
 x ) >.  e.  F  ->  E. y <. y ,  ( F `
 x ) >.  e.  F )
116, 10syl 14 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  E. y <. y ,  ( F `  x )
>.  e.  F )
12 funfvex 5322 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
13 elrn2g 4626 . . . . 5  |-  ( ( F `  x )  e.  _V  ->  (
( F `  x
)  e.  ran  F  <->  E. y <. y ,  ( F `  x )
>.  e.  F ) )
1412, 13syl 14 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  e.  ran  F  <->  E. y <. y ,  ( F `  x )
>.  e.  F ) )
1511, 14mpbird 165 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  ran  F
)
165, 15vtoclg 2679 . 2  |-  ( A  e.  dom  F  -> 
( ( Fun  F  /\  A  e.  dom  F )  ->  ( F `  A )  e.  ran  F ) )
1716anabsi7 548 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   <.cop 3449   dom cdm 4438   ran crn 4439   Fun wfun 5009   ` 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-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:  fnfvelrn  5431  eldmrexrn  5440  funfvima  5526  elunirn  5545  frecuzrdgdomlem  9824  frecuzrdgsuctlem  9830
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