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Theorem dfdfat2 38778
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 38762 . 2  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
2 relres 5138 . . . 4  |-  Rel  ( F  |`  { A }
)
3 dffun8 5616 . . . 4  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x  e.  dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y ) )
42, 3mpbiran 932 . . 3  |-  ( Fun  ( F  |`  { A } )  <->  A. x  e.  dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y )
54anbi2i 708 . 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 3034 . . . . . . . 8  |-  y  e. 
_V
76brres 5117 . . . . . . 7  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
87a1i 11 . . . . . 6  |-  ( A  e.  dom  F  -> 
( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e. 
{ A } ) ) )
98eubidv 2339 . . . . 5  |-  ( A  e.  dom  F  -> 
( E! y  x ( F  |`  { A } ) y  <->  E! y
( x F y  /\  x  e.  { A } ) ) )
109ralbidv 2829 . . . 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 5150 . . . . 5  |-  ( A  e.  dom  F  ->  A  e.  dom  ( F  |`  { A } ) )
12 eldmressn 38769 . . . . 5  |-  ( x  e.  dom  ( F  |`  { A } )  ->  x  =  A )
13 breq1 4398 . . . . . . . 8  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
1413anbi1d 719 . . . . . . 7  |-  ( x  =  A  ->  (
( x F y  /\  x  e.  { A } )  <->  ( A F y  /\  x  e.  { A } ) ) )
15 elsn 3973 . . . . . . . . 9  |-  ( x  e.  { A }  <->  x  =  A )
1615biimpri 211 . . . . . . . 8  |-  ( x  =  A  ->  x  e.  { A } )
1716biantrud 515 . . . . . . 7  |-  ( x  =  A  ->  ( A F y  <->  ( A F y  /\  x  e.  { A } ) ) )
1814, 17bitr4d 264 . . . . . 6  |-  ( x  =  A  ->  (
( x F y  /\  x  e.  { A } )  <->  A F
y ) )
1918eubidv 2339 . . . . 5  |-  ( x  =  A  ->  ( E! y ( x F y  /\  x  e. 
{ A } )  <-> 
E! y  A F y ) )
2011, 12, 19ralbinrald 38765 . . . 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 261 . . 3  |-  ( A  e.  dom  F  -> 
( A. x  e. 
dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y  <->  E! y  A F y ) )
2221pm5.32i 649 . 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 279 1  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  E! y  A F y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   E!weu 2319   A.wral 2756   {csn 3959   class class class wbr 4395   dom cdm 4839    |` cres 4841   Rel wrel 4844   Fun wfun 5583   defAt wdfat 38759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-res 4851  df-fun 5591  df-dfat 38762
This theorem is referenced by:  afveu  38800  rlimdmafv  38824
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