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

Theorem fniinfv 5858
Description: The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)
Assertion
Ref Expression
fniinfv  |-  ( F  Fn  A  ->  |^|_ x  e.  A  ( F `  x )  =  |^| ran 
F )
Distinct variable groups:    x, A    x, F

Proof of Theorem fniinfv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fnrnfv 5846 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
21inteqd 4240 . 2  |-  ( F  Fn  A  ->  |^| ran  F  =  |^| { y  |  E. x  e.  A  y  =  ( F `  x ) } )
3 fvex 5808 . . 3  |-  ( F `
 x )  e. 
_V
43dfiin2 4312 . 2  |-  |^|_ x  e.  A  ( F `  x )  =  |^| { y  |  E. x  e.  A  y  =  ( F `  x ) }
52, 4syl6reqr 2514 1  |-  ( F  Fn  A  ->  |^|_ x  e.  A  ( F `  x )  =  |^| ran 
F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   {cab 2439   E.wrex 2799   |^|cint 4235   |^|_ciin 4279   ran crn 4948    Fn wfn 5520   ` cfv 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-int 4236  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-iota 5488  df-fun 5527  df-fn 5528  df-fv 5533
This theorem is referenced by:  firest  14489  pnrmopn  19078  txtube  19344  bcth3  20973  imaiinfv  29176  diaintclN  35026  dibintclN  35135  dihintcl  35312
  Copyright terms: Public domain W3C validator