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Theorem afveu 27884
Description: The value of a function at a unique point, analogous to fveu 5679. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
Assertion
Ref Expression
afveu  |-  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
)
Distinct variable groups:    x, A    x, F

Proof of Theorem afveu
StepHypRef Expression
1 df-br 4173 . . . 4  |-  ( A F x  <->  <. A ,  x >.  e.  F )
21eubii 2263 . . 3  |-  ( E! x  A F x  <-> 
E! x <. A ,  x >.  e.  F )
3 eu2ndop1stv 27847 . . 3  |-  ( E! x <. A ,  x >.  e.  F  ->  A  e.  _V )
42, 3sylbi 188 . 2  |-  ( E! x  A F x  ->  A  e.  _V )
5 euex 2277 . . . . 5  |-  ( E! x  A F x  ->  E. x  A F x )
6 eldmg 5024 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  dom  F  <->  E. x  A F x ) )
75, 6syl5ibrcom 214 . . . 4  |-  ( E! x  A F x  ->  ( A  e. 
_V  ->  A  e.  dom  F ) )
87impcom 420 . . 3  |-  ( ( A  e.  _V  /\  E! x  A F x )  ->  A  e.  dom  F )
9 dfdfat2 27862 . . . . . . 7  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  E! x  A F x ) )
10 afvfundmfveq 27869 . . . . . . . . 9  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
11 fveu 5679 . . . . . . . . 9  |-  ( E! x  A F x  ->  ( F `  A )  =  U. { x  |  A F x } )
1210, 11sylan9eq 2456 . . . . . . . 8  |-  ( ( F defAt  A  /\  E! x  A F x )  ->  ( F''' A )  =  U. { x  |  A F x }
)
1312ex 424 . . . . . . 7  |-  ( F defAt 
A  ->  ( E! x  A F x  -> 
( F''' A )  =  U. { x  |  A F x } ) )
149, 13sylbir 205 . . . . . 6  |-  ( ( A  e.  dom  F  /\  E! x  A F x )  ->  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
) )
1514expcom 425 . . . . 5  |-  ( E! x  A F x  ->  ( A  e. 
dom  F  ->  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
) ) )
1615pm2.43a 47 . . . 4  |-  ( E! x  A F x  ->  ( A  e. 
dom  F  ->  ( F''' A )  =  U. { x  |  A F x } ) )
1716adantl 453 . . 3  |-  ( ( A  e.  _V  /\  E! x  A F x )  ->  ( A  e.  dom  F  -> 
( F''' A )  =  U. { x  |  A F x } ) )
188, 17mpd 15 . 2  |-  ( ( A  e.  _V  /\  E! x  A F x )  ->  ( F''' A )  =  U. { x  |  A F x } )
194, 18mpancom 651 1  |-  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   E!weu 2254   {cab 2390   _Vcvv 2916   <.cop 3777   U.cuni 3975   class class class wbr 4172   dom cdm 4837   ` cfv 5413   defAt wdfat 27838  '''cafv 27839
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-res 4849  df-iota 5377  df-fun 5415  df-fv 5421  df-dfat 27841  df-afv 27842
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