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Theorem dffv3 5798
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 3081 . . . . 5  |-  x  e. 
_V
2 elimasng 5306 . . . . . 6  |-  ( ( A  e.  _V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
3 df-br 4404 . . . . . 6  |-  ( A F x  <->  <. A ,  x >.  e.  F )
42, 3syl6bbr 263 . . . . 5  |-  ( ( A  e.  _V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  A F x ) )
51, 4mpan2 671 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  ( F
" { A }
)  <->  A F x ) )
65iotabidv 5513 . . 3  |-  ( A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x A F x ) )
7 df-fv 5537 . . 3  |-  ( F `
 A )  =  ( iota x A F x )
86, 7syl6reqr 2514 . 2  |-  ( A  e.  _V  ->  ( F `  A )  =  ( iota x x  e.  ( F " { A } ) ) )
9 fvprc 5796 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
10 snprc 4050 . . . . . . . . 9  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 194 . . . . . . . 8  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211imaeq2d 5280 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  ( F " (/) ) )
13 ima0 5295 . . . . . . 7  |-  ( F
" (/) )  =  (/)
1412, 13syl6eq 2511 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  (/) )
1514eleq2d 2524 . . . . 5  |-  ( -.  A  e.  _V  ->  ( x  e.  ( F
" { A }
)  <->  x  e.  (/) ) )
1615iotabidv 5513 . . . 4  |-  ( -.  A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x x  e.  (/) ) )
17 noel 3752 . . . . . . 7  |-  -.  x  e.  (/)
1817nex 1601 . . . . . 6  |-  -.  E. x  x  e.  (/)
19 euex 2290 . . . . . 6  |-  ( E! x  x  e.  (/)  ->  E. x  x  e.  (/) )
2018, 19mto 176 . . . . 5  |-  -.  E! x  x  e.  (/)
21 iotanul 5507 . . . . 5  |-  ( -.  E! x  x  e.  (/)  ->  ( iota x x  e.  (/) )  =  (/) )
2220, 21ax-mp 5 . . . 4  |-  ( iota
x x  e.  (/) )  =  (/)
2316, 22syl6eq 2511 . . 3  |-  ( -.  A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  (/) )
249, 23eqtr4d 2498 . 2  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  ( iota x x  e.  ( F " { A } ) ) )
258, 24pm2.61i 164 1  |-  ( F `
 A )  =  ( iota x x  e.  ( F " { A } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   E!weu 2262   _Vcvv 3078   (/)c0 3748   {csn 3988   <.cop 3994   class class class wbr 4403   "cima 4954   iotacio 5490   ` cfv 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-cnv 4959  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fv 5537
This theorem is referenced by:  dffv4  5799  fvco2  5878  shftval  12684  dffv5  28119
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