MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvelrn Structured version   Unicode version

Theorem fvelrn 5969
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 2489 . . . . 5  |-  ( x  =  A  ->  (
x  e.  dom  F  <->  A  e.  dom  F ) )
21anbi2d 708 . . . 4  |-  ( x  =  A  ->  (
( Fun  F  /\  x  e.  dom  F )  <-> 
( Fun  F  /\  A  e.  dom  F ) ) )
3 fveq2 5820 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
43eleq1d 2485 . . . 4  |-  ( x  =  A  ->  (
( F `  x
)  e.  ran  F  <->  ( F `  A )  e.  ran  F ) )
52, 4imbi12d 321 . . 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 5948 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. x ,  ( F `
 x ) >.  e.  F )
7 vex 3020 . . . . . 6  |-  x  e. 
_V
8 opeq1 4125 . . . . . . 7  |-  ( y  =  x  ->  <. y ,  ( F `  x ) >.  =  <. x ,  ( F `  x ) >. )
98eleq1d 2485 . . . . . 6  |-  ( y  =  x  ->  ( <. y ,  ( F `
 x ) >.  e.  F  <->  <. x ,  ( F `  x )
>.  e.  F ) )
107, 9spcev 3111 . . . . 5  |-  ( <.
x ,  ( F `
 x ) >.  e.  F  ->  E. y <. y ,  ( F `
 x ) >.  e.  F )
116, 10syl 17 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  E. y <. y ,  ( F `  x )
>.  e.  F )
12 fvex 5830 . . . . 5  |-  ( F `
 x )  e. 
_V
1312elrn2 5031 . . . 4  |-  ( ( F `  x )  e.  ran  F  <->  E. y <. y ,  ( F `
 x ) >.  e.  F )
1411, 13sylibr 215 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  ran  F
)
155, 14vtoclg 3077 . 2  |-  ( A  e.  dom  F  -> 
( ( Fun  F  /\  A  e.  dom  F )  ->  ( F `  A )  e.  ran  F ) )
1615anabsi7 826 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   <.cop 3942   dom cdm 4791   ran crn 4792   Fun wfun 5533   ` cfv 5539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pr 4598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-rab 2718  df-v 3019  df-sbc 3238  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4158  df-br 4362  df-opab 4421  df-id 4706  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-iota 5503  df-fun 5541  df-fn 5542  df-fv 5547
This theorem is referenced by:  nelrnfvne  5970  fnfvelrn  5973  eldmrexrn  5982  fvn0fvelrn  6035  funfvima  6094  elunirn  6110  rankwflemb  8211  dfac9  8512  fin1a2lem6  8781  gsumpropd2lem  16454  usgraedg3  25050  nbgraf1olem5  25110  usgrwlknloop  25230  usgra2wlkspthlem2  25285  nvnencycllem  25308  wlkiswwlk1  25355  usg2wlkonot  25548  usg2wotspth  25549  opfv  28188  nofv  30490  sltres  30497  bdayelon  30513  nodenselem3  30516  bj-elccinfty  31563  bj-minftyccb  31574  icoreunrn  31669  indexdom  31968  diaclN  34530  dia1elN  34534  docaclN  34604  dibclN  34642  dfac21  35837  cncmpmax  37269  icccncfext  37648  stoweidlem27  37770  stoweidlem29  37772  stoweidlem29OLD  37773  stoweidlem59  37803  fourierdlem20  37872  fourierdlem63  37916  fourierdlem76  37929  fourierdlem82  37935  fourierdlem93  37946  fourierdlem113  37966  fge0iccico  38063  sge0sn  38072  sge0tsms  38073  sge0cl  38074  sge0isum  38120  hoicvr  38217  afvelrn  38483  usgredg3  39043  usgredgedga  39054  subgruhgredgd  39093  suppdm  39897
  Copyright terms: Public domain W3C validator