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Theorem dffv3 5604
Description: A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)
Assertion
Ref Expression
dffv3  |-  ( F `
 A )  =  ( iota x x  e.  ( F " { A } ) )
Distinct variable groups:    x, F    x, A

Proof of Theorem dffv3
StepHypRef Expression
1 vex 2867 . . . . 5  |-  x  e. 
_V
2 elimasng 5121 . . . . . 6  |-  ( ( A  e.  _V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
3 df-br 4105 . . . . . 6  |-  ( A F x  <->  <. A ,  x >.  e.  F )
42, 3syl6bbr 254 . . . . 5  |-  ( ( A  e.  _V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  A F x ) )
51, 4mpan2 652 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  ( F
" { A }
)  <->  A F x ) )
65iotabidv 5322 . . 3  |-  ( A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x A F x ) )
7 df-fv 5345 . . 3  |-  ( F `
 A )  =  ( iota x A F x )
86, 7syl6reqr 2409 . 2  |-  ( A  e.  _V  ->  ( F `  A )  =  ( iota x x  e.  ( F " { A } ) ) )
9 fvprc 5602 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
10 snprc 3771 . . . . . . . . 9  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 186 . . . . . . . 8  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211imaeq2d 5094 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  ( F " (/) ) )
13 ima0 5112 . . . . . . 7  |-  ( F
" (/) )  =  (/)
1412, 13syl6eq 2406 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  (/) )
1514eleq2d 2425 . . . . 5  |-  ( -.  A  e.  _V  ->  ( x  e.  ( F
" { A }
)  <->  x  e.  (/) ) )
1615iotabidv 5322 . . . 4  |-  ( -.  A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x x  e.  (/) ) )
17 noel 3535 . . . . . . 7  |-  -.  x  e.  (/)
1817nex 1555 . . . . . 6  |-  -.  E. x  x  e.  (/)
19 euex 2232 . . . . . 6  |-  ( E! x  x  e.  (/)  ->  E. x  x  e.  (/) )
2018, 19mto 167 . . . . 5  |-  -.  E! x  x  e.  (/)
21 iotanul 5316 . . . . 5  |-  ( -.  E! x  x  e.  (/)  ->  ( iota x x  e.  (/) )  =  (/) )
2220, 21ax-mp 8 . . . 4  |-  ( iota
x x  e.  (/) )  =  (/)
2316, 22syl6eq 2406 . . 3  |-  ( -.  A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  (/) )
249, 23eqtr4d 2393 . 2  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  ( iota x x  e.  ( F " { A } ) ) )
258, 24pm2.61i 156 1  |-  ( F `
 A )  =  ( iota x x  e.  ( F " { A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358   E.wex 1541    = wceq 1642    e. wcel 1710   E!weu 2209   _Vcvv 2864   (/)c0 3531   {csn 3716   <.cop 3719   class class class wbr 4104   "cima 4774   iotacio 5299   ` cfv 5337
This theorem is referenced by:  dffv4  5605  fvco2  5677  shftval  11665  dffv5  25021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-xp 4777  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fv 5345
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