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Theorem fniinfv 5939
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 5925 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
21inteqd 4231 . 2  |-  ( F  Fn  A  ->  |^| ran  F  =  |^| { y  |  E. x  e.  A  y  =  ( F `  x ) } )
3 fvex 5889 . . 3  |-  ( F `
 x )  e. 
_V
43dfiin2 4304 . 2  |-  |^|_ x  e.  A  ( F `  x )  =  |^| { y  |  E. x  e.  A  y  =  ( F `  x ) }
52, 4syl6reqr 2524 1  |-  ( F  Fn  A  ->  |^|_ x  e.  A  ( F `  x )  =  |^| ran 
F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452   {cab 2457   E.wrex 2757   |^|cint 4226   |^|_ciin 4270   ran crn 4840    Fn wfn 5584   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597
This theorem is referenced by:  firest  15409  pnrmopn  20436  txtube  20732  bcth3  22377  diaintclN  34697  dibintclN  34806  dihintcl  34983  imaiinfv  35606
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