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Theorem afveu 28121
Description: The value of a function at a unique point, analogous to fveu 5533. (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 4040 . . . 4  |-  ( A F x  <->  <. A ,  x >.  e.  F )
21eubii 2165 . . 3  |-  ( E! x  A F x  <-> 
E! x <. A ,  x >.  e.  F )
3 eu2ndop1stv 28083 . . 3  |-  ( E! x <. A ,  x >.  e.  F  ->  A  e.  _V )
42, 3sylbi 187 . 2  |-  ( E! x  A F x  ->  A  e.  _V )
5 euex 2179 . . . . 5  |-  ( E! x  A F x  ->  E. x  A F x )
6 eldmg 4890 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  dom  F  <->  E. x  A F x ) )
75, 6syl5ibrcom 213 . . . 4  |-  ( E! x  A F x  ->  ( A  e. 
_V  ->  A  e.  dom  F ) )
87impcom 419 . . 3  |-  ( ( A  e.  _V  /\  E! x  A F x )  ->  A  e.  dom  F )
9 dfdfat2 28099 . . . . . . 7  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  E! x  A F x ) )
10 afvfundmfveq 28106 . . . . . . . . 9  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
11 fveu 5533 . . . . . . . . 9  |-  ( E! x  A F x  ->  ( F `  A )  =  U. { x  |  A F x } )
1210, 11sylan9eq 2348 . . . . . . . 8  |-  ( ( F defAt  A  /\  E! x  A F x )  ->  ( F''' A )  =  U. { x  |  A F x }
)
1312ex 423 . . . . . . 7  |-  ( F defAt 
A  ->  ( E! x  A F x  -> 
( F''' A )  =  U. { x  |  A F x } ) )
149, 13sylbir 204 . . . . . 6  |-  ( ( A  e.  dom  F  /\  E! x  A F x )  ->  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
) )
1514expcom 424 . . . . 5  |-  ( E! x  A F x  ->  ( A  e. 
dom  F  ->  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
) ) )
1615pm2.43a 45 . . . 4  |-  ( E! x  A F x  ->  ( A  e. 
dom  F  ->  ( F''' A )  =  U. { x  |  A F x } ) )
1716adantl 452 . . 3  |-  ( ( A  e.  _V  /\  E! x  A F x )  ->  ( A  e.  dom  F  -> 
( F''' A )  =  U. { x  |  A F x } ) )
188, 17mpd 14 . 2  |-  ( ( A  e.  _V  /\  E! x  A F x )  ->  ( F''' A )  =  U. { x  |  A F x } )
194, 18mpancom 650 1  |-  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   {cab 2282   _Vcvv 2801   <.cop 3656   U.cuni 3843   class class class wbr 4039   dom cdm 4705   ` cfv 5271   defAt wdfat 28074  '''cafv 28075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-dfat 28077  df-afv 28078
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