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Theorem dfdfat2 30037
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 30020 . 2  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
2 relres 5138 . . . 4  |-  Rel  ( F  |`  { A }
)
3 dffun8 5445 . . . 4  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x  e.  dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y ) )
42, 3mpbiran 909 . . 3  |-  ( Fun  ( F  |`  { A } )  <->  A. x  e.  dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y )
54anbi2i 694 . 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 2975 . . . . . . . 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 2276 . . . . 5  |-  ( A  e.  dom  F  -> 
( E! y  x ( F  |`  { A } ) y  <->  E! y
( x F y  /\  x  e.  { A } ) ) )
109ralbidv 2735 . . . 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 30029 . . . . 5  |-  ( A  e.  dom  F  ->  A  e.  dom  ( F  |`  { A } ) )
12 eldmressn 30027 . . . . 5  |-  ( x  e.  dom  ( F  |`  { A } )  ->  x  =  A )
13 breq1 4295 . . . . . . . 8  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
1413anbi1d 704 . . . . . . 7  |-  ( x  =  A  ->  (
( x F y  /\  x  e.  { A } )  <->  ( A F y  /\  x  e.  { A } ) ) )
15 elsn 3891 . . . . . . . . 9  |-  ( x  e.  { A }  <->  x  =  A )
1615biimpri 206 . . . . . . . 8  |-  ( x  =  A  ->  x  e.  { A } )
1716biantrud 507 . . . . . . 7  |-  ( x  =  A  ->  ( A F y  <->  ( A F y  /\  x  e.  { A } ) ) )
1814, 17bitr4d 256 . . . . . 6  |-  ( x  =  A  ->  (
( x F y  /\  x  e.  { A } )  <->  A F
y ) )
1918eubidv 2276 . . . . 5  |-  ( x  =  A  ->  ( E! y ( x F y  /\  x  e. 
{ A } )  <-> 
E! y  A F y ) )
2011, 12, 19ralbinrald 30023 . . . 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 253 . . 3  |-  ( A  e.  dom  F  -> 
( A. x  e. 
dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y  <->  E! y  A F y ) )
2221pm5.32i 637 . 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 271 1  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  E! y  A F y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E!weu 2253   A.wral 2715   {csn 3877   class class class wbr 4292   dom cdm 4840    |` cres 4842   Rel wrel 4845   Fun wfun 5412   defAt wdfat 30017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-res 4852  df-fun 5420  df-dfat 30020
This theorem is referenced by:  afveu  30059  rlimdmafv  30083
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