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Theorem afveu 31524
Description: The value of a function at a unique point, analogous to fveu 5849. (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 4441 . . . 4  |-  ( A F x  <->  <. A ,  x >.  e.  F )
21eubii 2293 . . 3  |-  ( E! x  A F x  <-> 
E! x <. A ,  x >.  e.  F )
3 eu2ndop1stv 31493 . . 3  |-  ( E! x <. A ,  x >.  e.  F  ->  A  e.  _V )
42, 3sylbi 195 . 2  |-  ( E! x  A F x  ->  A  e.  _V )
5 euex 2296 . . . . 5  |-  ( E! x  A F x  ->  E. x  A F x )
6 eldmg 5189 . . . . 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 31502 . . . . . . 7  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  E! x  A F x ) )
10 afvfundmfveq 31509 . . . . . . . . 9  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
11 fveu 5849 . . . . . . . . 9  |-  ( E! x  A F x  ->  ( F `  A )  =  U. { x  |  A F x } )
1210, 11sylan9eq 2521 . . . . . . . 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 1374   E.wex 1591    e. wcel 1762   E!weu 2268   {cab 2445   _Vcvv 3106   <.cop 4026   U.cuni 4238   class class class wbr 4440   dom cdm 4992   ` cfv 5579   defAt wdfat 31484  '''cafv 31485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-res 5004  df-iota 5542  df-fun 5581  df-fv 5587  df-dfat 31487  df-afv 31488
This theorem is referenced by: (None)
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