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Theorem fvelrn 5825
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 2464 . . . . 5  |-  ( x  =  A  ->  (
x  e.  dom  F  <->  A  e.  dom  F ) )
21anbi2d 685 . . . 4  |-  ( x  =  A  ->  (
( Fun  F  /\  x  e.  dom  F )  <-> 
( Fun  F  /\  A  e.  dom  F ) ) )
3 fveq2 5687 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
43eleq1d 2470 . . . 4  |-  ( x  =  A  ->  (
( F `  x
)  e.  ran  F  <->  ( F `  A )  e.  ran  F ) )
52, 4imbi12d 312 . . 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 5801 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. x ,  ( F `
 x ) >.  e.  F )
7 vex 2919 . . . . . 6  |-  x  e. 
_V
8 opeq1 3944 . . . . . . 7  |-  ( y  =  x  ->  <. y ,  ( F `  x ) >.  =  <. x ,  ( F `  x ) >. )
98eleq1d 2470 . . . . . 6  |-  ( y  =  x  ->  ( <. y ,  ( F `
 x ) >.  e.  F  <->  <. x ,  ( F `  x )
>.  e.  F ) )
107, 9spcev 3003 . . . . 5  |-  ( <.
x ,  ( F `
 x ) >.  e.  F  ->  E. y <. y ,  ( F `
 x ) >.  e.  F )
116, 10syl 16 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  E. y <. y ,  ( F `  x )
>.  e.  F )
12 fvex 5701 . . . . 5  |-  ( F `
 x )  e. 
_V
1312elrn2 5068 . . . 4  |-  ( ( F `  x )  e.  ran  F  <->  E. y <. y ,  ( F `
 x ) >.  e.  F )
1411, 13sylibr 204 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  ran  F
)
155, 14vtoclg 2971 . 2  |-  ( A  e.  dom  F  -> 
( ( Fun  F  /\  A  e.  dom  F )  ->  ( F `  A )  e.  ran  F ) )
1615anabsi7 793 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   <.cop 3777   dom cdm 4837   ran crn 4838   Fun wfun 5407   ` cfv 5413
This theorem is referenced by:  fnfvelrn  5826  eldmrexrn  5835  funfvima  5932  elunirnALT  5959  rankwflemb  7675  dfac9  7972  fin1a2lem6  8241  usgraedg3  21358  nbgraf1olem5  21408  usgrnloop  21516  nvnencycllem  21583  opfv  24011  gsumpropd2lem  24173  nofv  25525  sltres  25532  bdayelon  25548  nodenselem3  25551  indexdom  26326  dfac21  27032  cncmpmax  27570  stoweidlem27  27643  stoweidlem29  27645  stoweidlem59  27675  afvelrn  27899  usgra2wlkspthlem2  28037  usg2wlkonot  28080  usg2wotspth  28081  diaclN  31533  dia1elN  31537  docaclN  31607  dibclN  31645
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 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421
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