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Theorem fniinfv 5174
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  |^|_  F `

|^| ran  F
Distinct variable groups:   ,   , F

Proof of Theorem fniinfv
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 funfvex 5135 . . . . 5  Fun  F  dom  F  F `  _V
21funfni 4942 . . . 4  F  Fn  F `  _V
32ralrimiva 2386 . . 3  F  Fn  F `  _V
4 dfiin2g 3681 . . 3  F `  _V  |^|_  F `  |^| {  |  F `  }
53, 4syl 14 . 2  F  Fn  |^|_  F `

|^| {  |  F `  }
6 fnrnfv 5163 . . 3  F  Fn  ran  F  {  |  F `  }
76inteqd 3611 . 2  F  Fn  |^| ran 
F  |^| {  |  F `  }
85, 7eqtr4d 2072 1  F  Fn  |^|_  F `

|^| ran  F
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   wcel 1390   {cab 2023  wral 2300  wrex 2301   _Vcvv 2551   |^|cint 3606   |^|_ciin 3649   ran crn 4289    Fn wfn 4840   ` cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iin 3651  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by: (None)
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