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Theorem funresdfunsn 6106
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. (Contributed by AV, 2-Dec-2018.)
Assertion
Ref Expression
funresdfunsn  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( ( F  |`  ( _V  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } )  =  F )

Proof of Theorem funresdfunsn
StepHypRef Expression
1 funrel 5599 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
2 resdmdfsn 5150 . . . . 5  |-  ( Rel 
F  ->  ( F  |`  ( _V  \  { X } ) )  =  ( F  |`  ( dom  F  \  { X } ) ) )
31, 2syl 17 . . . 4  |-  ( Fun 
F  ->  ( F  |`  ( _V  \  { X } ) )  =  ( F  |`  ( dom  F  \  { X } ) ) )
43adantr 467 . . 3  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( F  |`  ( _V  \  { X }
) )  =  ( F  |`  ( dom  F 
\  { X }
) ) )
54uneq1d 3587 . 2  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( ( F  |`  ( _V  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } )  =  ( ( F  |`  ( dom  F  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
6 funfn 5611 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
7 fnsnsplit 6101 . . 3  |-  ( ( F  Fn  dom  F  /\  X  e.  dom  F )  ->  F  =  ( ( F  |`  ( dom  F  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
86, 7sylanb 475 . 2  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  F  =  ( ( F  |`  ( dom  F  \  { X } ) )  u.  { <. X ,  ( F `  X ) >. } ) )
95, 8eqtr4d 2488 1  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( ( F  |`  ( _V  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045    \ cdif 3401    u. cun 3402   {csn 3968   <.cop 3974   dom cdm 4834    |` cres 4836   Rel wrel 4839   Fun wfun 5576    Fn wfn 5577   ` cfv 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590
This theorem is referenced by:  setsidvald  15147
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