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Theorem dffv3g 5410
Description: A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
dffv3g  |-  ( A  e.  V  ->  ( F `  A )  =  ( iota x x  e.  ( F " { A } ) ) )
Distinct variable groups:    x, F    x, A    x, V

Proof of Theorem dffv3g
StepHypRef Expression
1 vex 2684 . . . 4  |-  x  e. 
_V
2 elimasng 4902 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
3 df-br 3925 . . . . 5  |-  ( A F x  <->  <. A ,  x >.  e.  F )
42, 3syl6bbr 197 . . . 4  |-  ( ( A  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  A F x ) )
51, 4mpan2 421 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( F
" { A }
)  <->  A F x ) )
65iotabidv 5104 . 2  |-  ( A  e.  V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x A F x ) )
7 df-fv 5126 . 2  |-  ( F `
 A )  =  ( iota x A F x )
86, 7syl6reqr 2189 1  |-  ( A  e.  V  ->  ( F `  A )  =  ( iota x x  e.  ( F " { A } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   _Vcvv 2681   {csn 3522   <.cop 3525   class class class wbr 3924   "cima 4537   iotacio 5081   ` cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fv 5126
This theorem is referenced by:  dffv4g  5411  fvco2  5483  shftval  10590
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