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Theorem fniinfv 5581
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 5569 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
21inteqd 3867 . 2  |-  ( F  Fn  A  ->  |^| ran  F  =  |^| { y  |  E. x  e.  A  y  =  ( F `  x ) } )
3 fvex 5539 . . 3  |-  ( F `
 x )  e. 
_V
43dfiin2 3938 . 2  |-  |^|_ x  e.  A  ( F `  x )  =  |^| { y  |  E. x  e.  A  y  =  ( F `  x ) }
52, 4syl6reqr 2334 1  |-  ( F  Fn  A  ->  |^|_ x  e.  A  ( F `  x )  =  |^| ran 
F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   {cab 2269   E.wrex 2544   |^|cint 3862   |^|_ciin 3906   ran crn 4690    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  firest  13337  pnrmopn  17071  txtube  17334  bcth3  18753  imaiinfv  26759  diaintclN  31248  dibintclN  31357  dihintcl  31534
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-int 3863  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  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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