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Theorem dffv3 5665
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 2903 . . . . 5  |-  x  e. 
_V
2 elimasng 5171 . . . . . 6  |-  ( ( A  e.  _V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
3 df-br 4155 . . . . . 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 5380 . . 3  |-  ( A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x A F x ) )
7 df-fv 5403 . . 3  |-  ( F `
 A )  =  ( iota x A F x )
86, 7syl6reqr 2439 . 2  |-  ( A  e.  _V  ->  ( F `  A )  =  ( iota x x  e.  ( F " { A } ) ) )
9 fvprc 5663 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
10 snprc 3815 . . . . . . . . 9  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 187 . . . . . . . 8  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211imaeq2d 5144 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  ( F " (/) ) )
13 ima0 5162 . . . . . . 7  |-  ( F
" (/) )  =  (/)
1412, 13syl6eq 2436 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  (/) )
1514eleq2d 2455 . . . . 5  |-  ( -.  A  e.  _V  ->  ( x  e.  ( F
" { A }
)  <->  x  e.  (/) ) )
1615iotabidv 5380 . . . 4  |-  ( -.  A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x x  e.  (/) ) )
17 noel 3576 . . . . . . 7  |-  -.  x  e.  (/)
1817nex 1561 . . . . . 6  |-  -.  E. x  x  e.  (/)
19 euex 2262 . . . . . 6  |-  ( E! x  x  e.  (/)  ->  E. x  x  e.  (/) )
2018, 19mto 169 . . . . 5  |-  -.  E! x  x  e.  (/)
21 iotanul 5374 . . . . 5  |-  ( -.  E! x  x  e.  (/)  ->  ( iota x x  e.  (/) )  =  (/) )
2220, 21ax-mp 8 . . . 4  |-  ( iota
x x  e.  (/) )  =  (/)
2316, 22syl6eq 2436 . . 3  |-  ( -.  A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  (/) )
249, 23eqtr4d 2423 . 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 1717   E!weu 2239   _Vcvv 2900   (/)c0 3572   {csn 3758   <.cop 3761   class class class wbr 4154   "cima 4822   iotacio 5357   ` cfv 5395
This theorem is referenced by:  dffv4  5666  fvco2  5738  shftval  11817  dffv5  25488
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-xp 4825  df-cnv 4827  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fv 5403
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