Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfdfat2 Structured version   Visualization version   Unicode version

Theorem dfdfat2 38627
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 38611 . 2  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
2 relres 5131 . . . 4  |-  Rel  ( F  |`  { A }
)
3 dffun8 5608 . . . 4  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x  e.  dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y ) )
42, 3mpbiran 928 . . 3  |-  ( Fun  ( F  |`  { A } )  <->  A. x  e.  dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y )
54anbi2i 699 . 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 3047 . . . . . . . 8  |-  y  e. 
_V
76brres 5110 . . . . . . 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 2318 . . . . 5  |-  ( A  e.  dom  F  -> 
( E! y  x ( F  |`  { A } ) y  <->  E! y
( x F y  /\  x  e.  { A } ) ) )
109ralbidv 2826 . . . 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 5143 . . . . 5  |-  ( A  e.  dom  F  ->  A  e.  dom  ( F  |`  { A } ) )
12 eldmressn 38618 . . . . 5  |-  ( x  e.  dom  ( F  |`  { A } )  ->  x  =  A )
13 breq1 4404 . . . . . . . 8  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
1413anbi1d 710 . . . . . . 7  |-  ( x  =  A  ->  (
( x F y  /\  x  e.  { A } )  <->  ( A F y  /\  x  e.  { A } ) ) )
15 elsn 3981 . . . . . . . . 9  |-  ( x  e.  { A }  <->  x  =  A )
1615biimpri 210 . . . . . . . 8  |-  ( x  =  A  ->  x  e.  { A } )
1716biantrud 510 . . . . . . 7  |-  ( x  =  A  ->  ( A F y  <->  ( A F y  /\  x  e.  { A } ) ) )
1814, 17bitr4d 260 . . . . . 6  |-  ( x  =  A  ->  (
( x F y  /\  x  e.  { A } )  <->  A F
y ) )
1918eubidv 2318 . . . . 5  |-  ( x  =  A  ->  ( E! y ( x F y  /\  x  e. 
{ A } )  <-> 
E! y  A F y ) )
2011, 12, 19ralbinrald 38614 . . . 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 257 . . 3  |-  ( A  e.  dom  F  -> 
( A. x  e. 
dom  ( F  |`  { A } ) E! y  x ( F  |`  { A } ) y  <->  E! y  A F y ) )
2221pm5.32i 642 . 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 275 1  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  E! y  A F y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886   E!weu 2298   A.wral 2736   {csn 3967   class class class wbr 4401   dom cdm 4833    |` cres 4835   Rel wrel 4838   Fun wfun 5575   defAt wdfat 38608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-opab 4461  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-res 4845  df-fun 5583  df-dfat 38611
This theorem is referenced by:  afveu  38649  rlimdmafv  38673
  Copyright terms: Public domain W3C validator