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Theorem funresdfunsndc 6305
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 5066 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
2 resdmdfsn 4788 . . . . 5  |-  ( Rel 
F  ->  ( F  |`  ( _V  \  { X } ) )  =  ( F  |`  ( dom  F  \  { X } ) ) )
31, 2syl 14 . . . 4  |-  ( Fun 
F  ->  ( F  |`  ( _V  \  { X } ) )  =  ( F  |`  ( dom  F  \  { X } ) ) )
433ad2ant2 968 . . 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 3168 . 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 5079 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
7 fnsnsplitdc 6304 . . 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 1218 . 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 2130 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 783    /\ w3a 927    = wceq 1296    e. wcel 1445   A.wral 2370   _Vcvv 2633    \ cdif 3010    u. cun 3011   {csn 3466   <.cop 3469   dom cdm 4467    |` cres 4469   Rel wrel 4472   Fun wfun 5043    Fn wfn 5044   ` cfv 5049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057
This theorem is referenced by:  strsetsid  11691
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