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Theorem fniinfv 5263
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 funfvex 5223 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
21funfni 5030 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
32ralrimiva 2435 . . 3  |-  ( F  Fn  A  ->  A. x  e.  A  ( F `  x )  e.  _V )
4 dfiin2g 3719 . . 3  |-  ( A. x  e.  A  ( F `  x )  e.  _V  ->  |^|_ x  e.  A  ( F `  x )  =  |^| { y  |  E. x  e.  A  y  =  ( F `  x ) } )
53, 4syl 14 . 2  |-  ( F  Fn  A  ->  |^|_ x  e.  A  ( F `  x )  =  |^| { y  |  E. x  e.  A  y  =  ( F `  x ) } )
6 fnrnfv 5252 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
76inteqd 3649 . 2  |-  ( F  Fn  A  ->  |^| ran  F  =  |^| { y  |  E. x  e.  A  y  =  ( F `  x ) } )
85, 7eqtr4d 2117 1  |-  ( F  Fn  A  ->  |^|_ x  e.  A  ( F `  x )  =  |^| ran 
F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   E.wrex 2350   _Vcvv 2602   |^|cint 3644   |^|_ciin 3687   ran crn 4372    Fn wfn 4927   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iin 3689  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940
This theorem is referenced by: (None)
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