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Theorem dffv3 5521
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 2791 . . . . 5  |-  x  e. 
_V
2 elimasng 5039 . . . . . 6  |-  ( ( A  e.  _V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
3 df-br 4024 . . . . . 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 5240 . . 3  |-  ( A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x A F x ) )
7 df-fv 5263 . . 3  |-  ( F `
 A )  =  ( iota x A F x )
86, 7syl6reqr 2334 . 2  |-  ( A  e.  _V  ->  ( F `  A )  =  ( iota x x  e.  ( F " { A } ) ) )
9 fvprc 5519 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
10 snprc 3695 . . . . . . . . 9  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 186 . . . . . . . 8  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211imaeq2d 5012 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  ( F " (/) ) )
13 ima0 5030 . . . . . . 7  |-  ( F
" (/) )  =  (/)
1412, 13syl6eq 2331 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  (/) )
1514eleq2d 2350 . . . . 5  |-  ( -.  A  e.  _V  ->  ( x  e.  ( F
" { A }
)  <->  x  e.  (/) ) )
1615iotabidv 5240 . . . 4  |-  ( -.  A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  ( iota x x  e.  (/) ) )
17 noel 3459 . . . . . . 7  |-  -.  x  e.  (/)
1817nex 1542 . . . . . 6  |-  -.  E. x  x  e.  (/)
19 euex 2166 . . . . . 6  |-  ( E! x  x  e.  (/)  ->  E. x  x  e.  (/) )
2018, 19mto 167 . . . . 5  |-  -.  E! x  x  e.  (/)
21 iotanul 5234 . . . . 5  |-  ( -.  E! x  x  e.  (/)  ->  ( iota x x  e.  (/) )  =  (/) )
2220, 21ax-mp 8 . . . 4  |-  ( iota
x x  e.  (/) )  =  (/)
2316, 22syl6eq 2331 . . 3  |-  ( -.  A  e.  _V  ->  ( iota x x  e.  ( F " { A } ) )  =  (/) )
249, 23eqtr4d 2318 . 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 1528    = wceq 1623    e. wcel 1684   E!weu 2143   _Vcvv 2788   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023   "cima 4692   iotacio 5217   ` cfv 5255
This theorem is referenced by:  dffv4  5522  fvco2  5594  shftval  11569  dffv5  24463
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263
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