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Theorem dffv3g 5463
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 5177 . 2  |-  ( F `
 A )  =  ( iota x A F x )
2 vex 2715 . . . 4  |-  x  e. 
_V
3 elimasng 4953 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
4 df-br 3966 . . . . 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 422 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( F
" { A }
)  <->  A F x ) )
76iotabidv 5155 . 2  |-  ( A  e.  V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x A F x ) )
81, 7eqtr4id 2209 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 1335    e. wcel 2128   _Vcvv 2712   {csn 3560   <.cop 3563   class class class wbr 3965   "cima 4588   iotacio 5132   ` cfv 5169
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4591  df-cnv 4593  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fv 5177
This theorem is referenced by:  dffv4g  5464  fvco2  5536  shftval  10720
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