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Theorem fniinfv 5929
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 5916 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
21inteqd 4242 . 2  |-  ( F  Fn  A  ->  |^| ran  F  =  |^| { y  |  E. x  e.  A  y  =  ( F `  x ) } )
3 fvex 5880 . . 3  |-  ( F `
 x )  e. 
_V
43dfiin2 4316 . 2  |-  |^|_ x  e.  A  ( F `  x )  =  |^| { y  |  E. x  e.  A  y  =  ( F `  x ) }
52, 4syl6reqr 2506 1  |-  ( F  Fn  A  ->  |^|_ x  e.  A  ( F `  x )  =  |^| ran 
F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1446   {cab 2439   E.wrex 2740   |^|cint 4237   |^|_ciin 4282   ran crn 4838    Fn wfn 5580   ` cfv 5585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-int 4238  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5549  df-fun 5587  df-fn 5588  df-fv 5593
This theorem is referenced by:  firest  15343  pnrmopn  20371  txtube  20667  bcth3  22311  diaintclN  34638  dibintclN  34747  dihintcl  34924  imaiinfv  35547
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