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Theorem fvelrn 5661
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 2343 . . . . 5  |-  ( x  =  A  ->  (
x  e.  dom  F  <->  A  e.  dom  F ) )
21anbi2d 684 . . . 4  |-  ( x  =  A  ->  (
( Fun  F  /\  x  e.  dom  F )  <-> 
( Fun  F  /\  A  e.  dom  F ) ) )
3 fveq2 5525 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
43eleq1d 2349 . . . 4  |-  ( x  =  A  ->  (
( F `  x
)  e.  ran  F  <->  ( F `  A )  e.  ran  F ) )
52, 4imbi12d 311 . . 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 5637 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. x ,  ( F `
 x ) >.  e.  F )
7 vex 2791 . . . . . 6  |-  x  e. 
_V
8 opeq1 3796 . . . . . . 7  |-  ( y  =  x  ->  <. y ,  ( F `  x ) >.  =  <. x ,  ( F `  x ) >. )
98eleq1d 2349 . . . . . 6  |-  ( y  =  x  ->  ( <. y ,  ( F `
 x ) >.  e.  F  <->  <. x ,  ( F `  x )
>.  e.  F ) )
107, 9spcev 2875 . . . . 5  |-  ( <.
x ,  ( F `
 x ) >.  e.  F  ->  E. y <. y ,  ( F `
 x ) >.  e.  F )
116, 10syl 15 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  E. y <. y ,  ( F `  x )
>.  e.  F )
12 fvex 5539 . . . . 5  |-  ( F `
 x )  e. 
_V
1312elrn2 4918 . . . 4  |-  ( ( F `  x )  e.  ran  F  <->  E. y <. y ,  ( F `
 x ) >.  e.  F )
1411, 13sylibr 203 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  ran  F
)
155, 14vtoclg 2843 . 2  |-  ( A  e.  dom  F  -> 
( ( Fun  F  /\  A  e.  dom  F )  ->  ( F `  A )  e.  ran  F ) )
1615anabsi7 792 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   <.cop 3643   dom cdm 4689   ran crn 4690   Fun wfun 5249   ` cfv 5255
This theorem is referenced by:  fnfvelrn  5662  funfvima  5753  elunirnALT  5779  rankwflemb  7465  dfac9  7762  fin1a2lem6  8031  opfv  23190  gsumpropd2lem  23379  nofv  24311  sltres  24318  bdayelon  24334  nodenselem3  24337  fnovrn2  25050  bwt2  25592  supnuf  25629  rdmob  25748  rcmob  25749  pfsubkl  26047  indexdom  26413  dfac21  27164  cncmpmax  27703  stoweidlem27  27776  stoweidlem29  27778  stoweidlem59  27808  afvelrn  28030  diaclN  31240  dia1elN  31244  docaclN  31314  dibclN  31352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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