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Theorem dfdfat2 28099
Description: Alternate definition of the predicate "defined at" not using the  Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
Assertion
Ref Expression
dfdfat2  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  E! y  A F y ) )
Distinct variable groups:    y, A    y, F

Proof of Theorem dfdfat2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-dfat 28077 . 2  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
2 relres 4999 . . . 4  |-  Rel  ( F  |`  { A }
)
3 dffun8 5297 . . . 4  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x  e.  dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y ) )
42, 3mpbiran 884 . . 3  |-  ( Fun  ( F  |`  { A } )  <->  A. x  e.  dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y )
54anbi2i 675 . 2  |-  ( ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  <-> 
( A  e.  dom  F  /\  A. x  e. 
dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y ) )
6 vex 2804 . . . . . . . 8  |-  y  e. 
_V
76brres 4977 . . . . . . 7  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
87a1i 10 . . . . . 6  |-  ( A  e.  dom  F  -> 
( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e. 
{ A } ) ) )
98eubidv 2164 . . . . 5  |-  ( A  e.  dom  F  -> 
( E! y  x ( F  |`  { A } ) y  <->  E! y
( x F y  /\  x  e.  { A } ) ) )
109ralbidv 2576 . . . 4  |-  ( A  e.  dom  F  -> 
( A. x  e. 
dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y  <->  A. x  e.  dom  ( F  |`  { A } ) E! y ( x F y  /\  x  e.  { A } ) ) )
11 eldmressnsn 28089 . . . . 5  |-  ( A  e.  dom  F  ->  A  e.  dom  ( F  |`  { A } ) )
12 eldmressn 28087 . . . . 5  |-  ( x  e.  dom  ( F  |`  { A } )  ->  x  =  A )
13 breq1 4042 . . . . . . . 8  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
1413anbi1d 685 . . . . . . 7  |-  ( x  =  A  ->  (
( x F y  /\  x  e.  { A } )  <->  ( A F y  /\  x  e.  { A } ) ) )
15 elsn 3668 . . . . . . . . 9  |-  ( x  e.  { A }  <->  x  =  A )
1615biimpri 197 . . . . . . . 8  |-  ( x  =  A  ->  x  e.  { A } )
1716biantrud 493 . . . . . . 7  |-  ( x  =  A  ->  ( A F y  <->  ( A F y  /\  x  e.  { A } ) ) )
1814, 17bitr4d 247 . . . . . 6  |-  ( x  =  A  ->  (
( x F y  /\  x  e.  { A } )  <->  A F
y ) )
1918eubidv 2164 . . . . 5  |-  ( x  =  A  ->  ( E! y ( x F y  /\  x  e. 
{ A } )  <-> 
E! y  A F y ) )
2011, 12, 19ralbinrald 28080 . . . 4  |-  ( A  e.  dom  F  -> 
( A. x  e. 
dom  ( F  |`  { A } ) E! y ( x F y  /\  x  e. 
{ A } )  <-> 
E! y  A F y ) )
2110, 20bitrd 244 . . 3  |-  ( A  e.  dom  F  -> 
( A. x  e. 
dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y  <->  E! y  A F y ) )
2221pm5.32i 618 . 2  |-  ( ( A  e.  dom  F  /\  A. x  e.  dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y )  <->  ( A  e. 
dom  F  /\  E! y  A F y ) )
231, 5, 223bitri 262 1  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  E! y  A F y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E!weu 2156   A.wral 2556   {csn 3653   class class class wbr 4039   dom cdm 4705    |` cres 4707   Rel wrel 4710   Fun wfun 5265   defAt wdfat 28074
This theorem is referenced by:  afveu  28121  rlimdmafv  28145
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  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-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-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-fun 5273  df-dfat 28077
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