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Theorem dffv3g 5285
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 2622 . . . 4  |-  x  e. 
_V
2 elimasng 4787 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
3 df-br 3838 . . . . 5  |-  ( A F x  <->  <. A ,  x >.  e.  F )
42, 3syl6bbr 196 . . . 4  |-  ( ( A  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  A F x ) )
51, 4mpan2 416 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( F
" { A }
)  <->  A F x ) )
65iotabidv 4988 . 2  |-  ( A  e.  V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x A F x ) )
7 df-fv 5010 . 2  |-  ( F `
 A )  =  ( iota x A F x )
86, 7syl6reqr 2139 1  |-  ( A  e.  V  ->  ( F `  A )  =  ( iota x x  e.  ( F " { A } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619   {csn 3441   <.cop 3444   class class class wbr 3837   "cima 4431   iotacio 4965   ` cfv 5002
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fv 5010
This theorem is referenced by:  dffv4g  5286  fvco2  5357  shftval  10224
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