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Theorem dffv3g 5492
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 df-fv 5206 . 2  |-  ( F `
 A )  =  ( iota x A F x )
2 vex 2733 . . . 4  |-  x  e. 
_V
3 elimasng 4979 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
4 df-br 3990 . . . . 5  |-  ( A F x  <->  <. A ,  x >.  e.  F )
53, 4bitr4di 197 . . . 4  |-  ( ( A  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  A F x ) )
62, 5mpan2 423 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( F
" { A }
)  <->  A F x ) )
76iotabidv 5181 . 2  |-  ( A  e.  V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x A F x ) )
81, 7eqtr4id 2222 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 1348    e. wcel 2141   _Vcvv 2730   {csn 3583   <.cop 3586   class class class wbr 3989   "cima 4614   iotacio 5158   ` cfv 5198
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fv 5206
This theorem is referenced by:  dffv4g  5493  fvco2  5565  shftval  10789
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