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Theorem funresdfunsn 5919
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 5434 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
2 resdmdfsn 5151 . . . . 5  |-  ( Rel 
F  ->  ( F  |`  ( _V  \  { X } ) )  =  ( F  |`  ( dom  F  \  { X } ) ) )
31, 2syl 16 . . . 4  |-  ( Fun 
F  ->  ( F  |`  ( _V  \  { X } ) )  =  ( F  |`  ( dom  F  \  { X } ) ) )
43adantr 465 . . 3  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( F  |`  ( _V  \  { X }
) )  =  ( F  |`  ( dom  F 
\  { X }
) ) )
54uneq1d 3508 . 2  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( ( F  |`  ( _V  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } )  =  ( ( F  |`  ( dom  F  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
6 funfn 5446 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
7 fnsnsplit 5914 . . 3  |-  ( ( F  Fn  dom  F  /\  X  e.  dom  F )  ->  F  =  ( ( F  |`  ( dom  F  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
86, 7sylanb 472 . 2  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  F  =  ( ( F  |`  ( dom  F  \  { X } ) )  u.  { <. X ,  ( F `  X ) >. } ) )
95, 8eqtr4d 2477 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 369    = wceq 1369    e. wcel 1756   _Vcvv 2971    \ cdif 3324    u. cun 3325   {csn 3876   <.cop 3882   dom cdm 4839    |` cres 4841   Rel wrel 4844   Fun wfun 5411    Fn wfn 5412   ` cfv 5417
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 4412  ax-nul 4420  ax-pr 4530
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425
This theorem is referenced by: (None)
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