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Theorem dffv3 5691
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 2927 . . . . 5  |-  x  e. 
_V
2 elimasng 5197 . . . . . 6  |-  ( ( A  e.  _V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
3 df-br 4181 . . . . . 6  |-  ( A F x  <->  <. A ,  x >.  e.  F )
42, 3syl6bbr 255 . . . . 5  |-  ( ( A  e.  _V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  A F x ) )
51, 4mpan2 653 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  ( F
" { A }
)  <->  A F x ) )
65iotabidv 5406 . . 3  |-  ( A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x A F x ) )
7 df-fv 5429 . . 3  |-  ( F `
 A )  =  ( iota x A F x )
86, 7syl6reqr 2463 . 2  |-  ( A  e.  _V  ->  ( F `  A )  =  ( iota x x  e.  ( F " { A } ) ) )
9 fvprc 5689 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
10 snprc 3839 . . . . . . . . 9  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 187 . . . . . . . 8  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211imaeq2d 5170 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  ( F " (/) ) )
13 ima0 5188 . . . . . . 7  |-  ( F
" (/) )  =  (/)
1412, 13syl6eq 2460 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  (/) )
1514eleq2d 2479 . . . . 5  |-  ( -.  A  e.  _V  ->  ( x  e.  ( F
" { A }
)  <->  x  e.  (/) ) )
1615iotabidv 5406 . . . 4  |-  ( -.  A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x x  e.  (/) ) )
17 noel 3600 . . . . . . 7  |-  -.  x  e.  (/)
1817nex 1561 . . . . . 6  |-  -.  E. x  x  e.  (/)
19 euex 2285 . . . . . 6  |-  ( E! x  x  e.  (/)  ->  E. x  x  e.  (/) )
2018, 19mto 169 . . . . 5  |-  -.  E! x  x  e.  (/)
21 iotanul 5400 . . . . 5  |-  ( -.  E! x  x  e.  (/)  ->  ( iota x x  e.  (/) )  =  (/) )
2220, 21ax-mp 8 . . . 4  |-  ( iota
x x  e.  (/) )  =  (/)
2316, 22syl6eq 2460 . . 3  |-  ( -.  A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  (/) )
249, 23eqtr4d 2447 . 2  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  ( iota x x  e.  ( F " { A } ) ) )
258, 24pm2.61i 158 1  |-  ( F `
 A )  =  ( iota x x  e.  ( F " { A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   E!weu 2262   _Vcvv 2924   (/)c0 3596   {csn 3782   <.cop 3785   class class class wbr 4180   "cima 4848   iotacio 5383   ` cfv 5421
This theorem is referenced by:  dffv4  5692  fvco2  5765  shftval  11852  dffv5  25685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-xp 4851  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fv 5429
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