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Theorem fvelrn 5519
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 2180 . . . . 5  |-  ( x  =  A  ->  (
x  e.  dom  F  <->  A  e.  dom  F ) )
21anbi2d 459 . . . 4  |-  ( x  =  A  ->  (
( Fun  F  /\  x  e.  dom  F )  <-> 
( Fun  F  /\  A  e.  dom  F ) ) )
3 fveq2 5389 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
43eleq1d 2186 . . . 4  |-  ( x  =  A  ->  (
( F `  x
)  e.  ran  F  <->  ( F `  A )  e.  ran  F ) )
52, 4imbi12d 233 . . 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 5500 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. x ,  ( F `
 x ) >.  e.  F )
7 vex 2663 . . . . . 6  |-  x  e. 
_V
8 opeq1 3675 . . . . . . 7  |-  ( y  =  x  ->  <. y ,  ( F `  x ) >.  =  <. x ,  ( F `  x ) >. )
98eleq1d 2186 . . . . . 6  |-  ( y  =  x  ->  ( <. y ,  ( F `
 x ) >.  e.  F  <->  <. x ,  ( F `  x )
>.  e.  F ) )
107, 9spcev 2754 . . . . 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 5406 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
13 elrn2g 4699 . . . . 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 166 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  ran  F
)
165, 15vtoclg 2720 . 2  |-  ( A  e.  dom  F  -> 
( ( Fun  F  /\  A  e.  dom  F )  ->  ( F `  A )  e.  ran  F ) )
1716anabsi7 555 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465   _Vcvv 2660   <.cop 3500   dom cdm 4509   ran crn 4510   Fun wfun 5087   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  fnfvelrn  5520  eldmrexrn  5529  funfvima  5617  elunirn  5635  frecuzrdgdomlem  10145  frecuzrdgsuctlem  10151
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