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Theorem funresdfunsndc 6535
Description: Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself, where equality is decidable. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 30-Jan-2023.)
Assertion
Ref Expression
funresdfunsndc  |-  ( ( A. x  e.  dom  F A. y  e.  dom  FDECID  x  =  y  /\  Fun  F  /\  X  e.  dom  F )  ->  ( ( F  |`  ( _V  \  { X } ) )  u.  { <. X , 
( F `  X
) >. } )  =  F )
Distinct variable groups:    x, F, y   
x, X, y

Proof of Theorem funresdfunsndc
StepHypRef Expression
1 funrel 5255 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
2 resdmdfsn 4971 . . . . 5  |-  ( Rel 
F  ->  ( F  |`  ( _V  \  { X } ) )  =  ( F  |`  ( dom  F  \  { X } ) ) )
31, 2syl 14 . . . 4  |-  ( Fun 
F  ->  ( F  |`  ( _V  \  { X } ) )  =  ( F  |`  ( dom  F  \  { X } ) ) )
433ad2ant2 1021 . . 3  |-  ( ( A. x  e.  dom  F A. y  e.  dom  FDECID  x  =  y  /\  Fun  F  /\  X  e.  dom  F )  ->  ( F  |`  ( _V  \  { X } ) )  =  ( F  |`  ( dom  F  \  { X } ) ) )
54uneq1d 3303 . 2  |-  ( ( A. x  e.  dom  F A. y  e.  dom  FDECID  x  =  y  /\  Fun  F  /\  X  e.  dom  F )  ->  ( ( F  |`  ( _V  \  { X } ) )  u.  { <. X , 
( F `  X
) >. } )  =  ( ( F  |`  ( dom  F  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
6 funfn 5268 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
7 fnsnsplitdc 6534 . . 3  |-  ( ( A. x  e.  dom  F A. y  e.  dom  FDECID  x  =  y  /\  F  Fn  dom  F  /\  X  e.  dom  F )  ->  F  =  ( ( F  |`  ( dom  F  \  { X } ) )  u.  { <. X ,  ( F `  X ) >. } ) )
86, 7syl3an2b 1286 . 2  |-  ( ( A. x  e.  dom  F A. y  e.  dom  FDECID  x  =  y  /\  Fun  F  /\  X  e.  dom  F )  ->  F  =  ( ( F  |`  ( dom  F  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
95, 8eqtr4d 2225 1  |-  ( ( A. x  e.  dom  F A. y  e.  dom  FDECID  x  =  y  /\  Fun  F  /\  X  e.  dom  F )  ->  ( ( F  |`  ( _V  \  { X } ) )  u.  { <. X , 
( F `  X
) >. } )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4  DECID wdc 835    /\ w3a 980    = wceq 1364    e. wcel 2160   A.wral 2468   _Vcvv 2752    \ cdif 3141    u. cun 3142   {csn 3610   <.cop 3613   dom cdm 4647    |` cres 4649   Rel wrel 4652   Fun wfun 5232    Fn wfn 5233   ` cfv 5238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246
This theorem is referenced by:  strsetsid  12556
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