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Theorem afveu 32076
Description: The value of a function at a unique point, analogous to fveu 5848. (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 4438 . . . 4  |-  ( A F x  <->  <. A ,  x >.  e.  F )
21eubii 2292 . . 3  |-  ( E! x  A F x  <-> 
E! x <. A ,  x >.  e.  F )
3 eu2ndop1stv 32045 . . 3  |-  ( E! x <. A ,  x >.  e.  F  ->  A  e.  _V )
42, 3sylbi 195 . 2  |-  ( E! x  A F x  ->  A  e.  _V )
5 euex 2294 . . . . 5  |-  ( E! x  A F x  ->  E. x  A F x )
6 eldmg 5188 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  dom  F  <->  E. x  A F x ) )
75, 6syl5ibrcom 222 . . . 4  |-  ( E! x  A F x  ->  ( A  e. 
_V  ->  A  e.  dom  F ) )
87impcom 430 . . 3  |-  ( ( A  e.  _V  /\  E! x  A F x )  ->  A  e.  dom  F )
9 dfdfat2 32054 . . . . . . 7  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  E! x  A F x ) )
10 afvfundmfveq 32061 . . . . . . . . 9  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
11 fveu 5848 . . . . . . . . 9  |-  ( E! x  A F x  ->  ( F `  A )  =  U. { x  |  A F x } )
1210, 11sylan9eq 2504 . . . . . . . 8  |-  ( ( F defAt  A  /\  E! x  A F x )  ->  ( F''' A )  =  U. { x  |  A F x }
)
1312ex 434 . . . . . . 7  |-  ( F defAt 
A  ->  ( E! x  A F x  -> 
( F''' A )  =  U. { x  |  A F x } ) )
149, 13sylbir 213 . . . . . 6  |-  ( ( A  e.  dom  F  /\  E! x  A F x )  ->  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
) )
1514expcom 435 . . . . 5  |-  ( E! x  A F x  ->  ( A  e. 
dom  F  ->  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
) ) )
1615pm2.43a 49 . . . 4  |-  ( E! x  A F x  ->  ( A  e. 
dom  F  ->  ( F''' A )  =  U. { x  |  A F x } ) )
1716adantl 466 . . 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 669 1  |-  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804   E!weu 2268   {cab 2428   _Vcvv 3095   <.cop 4020   U.cuni 4234   class class class wbr 4437   dom cdm 4989   ` cfv 5578   defAt wdfat 32036  '''cafv 32037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-res 5001  df-iota 5541  df-fun 5580  df-fv 5586  df-dfat 32039  df-afv 32040
This theorem is referenced by: (None)
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