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Theorem dfdfat2 27971
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 27950 . 2  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
2 relres 5174 . . . 4  |-  Rel  ( F  |`  { A }
)
3 dffun8 5480 . . . 4  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x  e.  dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y ) )
42, 3mpbiran 885 . . 3  |-  ( Fun  ( F  |`  { A } )  <->  A. x  e.  dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y )
54anbi2i 676 . 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 2959 . . . . . . . 8  |-  y  e. 
_V
76brres 5152 . . . . . . 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 2289 . . . . 5  |-  ( A  e.  dom  F  -> 
( E! y  x ( F  |`  { A } ) y  <->  E! y
( x F y  /\  x  e.  { A } ) ) )
109ralbidv 2725 . . . 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 27962 . . . . 5  |-  ( A  e.  dom  F  ->  A  e.  dom  ( F  |`  { A } ) )
12 eldmressn 27960 . . . . 5  |-  ( x  e.  dom  ( F  |`  { A } )  ->  x  =  A )
13 breq1 4215 . . . . . . . 8  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
1413anbi1d 686 . . . . . . 7  |-  ( x  =  A  ->  (
( x F y  /\  x  e.  { A } )  <->  ( A F y  /\  x  e.  { A } ) ) )
15 elsn 3829 . . . . . . . . 9  |-  ( x  e.  { A }  <->  x  =  A )
1615biimpri 198 . . . . . . . 8  |-  ( x  =  A  ->  x  e.  { A } )
1716biantrud 494 . . . . . . 7  |-  ( x  =  A  ->  ( A F y  <->  ( A F y  /\  x  e.  { A } ) ) )
1814, 17bitr4d 248 . . . . . 6  |-  ( x  =  A  ->  (
( x F y  /\  x  e.  { A } )  <->  A F
y ) )
1918eubidv 2289 . . . . 5  |-  ( x  =  A  ->  ( E! y ( x F y  /\  x  e. 
{ A } )  <-> 
E! y  A F y ) )
2011, 12, 19ralbinrald 27953 . . . 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 245 . . 3  |-  ( A  e.  dom  F  -> 
( A. x  e. 
dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y  <->  E! y  A F y ) )
2221pm5.32i 619 . 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 263 1  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  E! y  A F y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E!weu 2281   A.wral 2705   {csn 3814   class class class wbr 4212   dom cdm 4878    |` cres 4880   Rel wrel 4883   Fun wfun 5448   defAt wdfat 27947
This theorem is referenced by:  afveu  27993  rlimdmafv  28017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-res 4890  df-fun 5456  df-dfat 27950
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