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Theorem funresdfunsn 6114
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 5611 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
2 resdmdfsn 5329 . . . . 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 3653 . 2  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( ( F  |`  ( _V  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } )  =  ( ( F  |`  ( dom  F  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
6 funfn 5623 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
7 fnsnsplit 6109 . . 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 2501 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 1395    e. wcel 1819   _Vcvv 3109    \ cdif 3468    u. cun 3469   {csn 4032   <.cop 4038   dom cdm 5008    |` cres 5010   Rel wrel 5013   Fun wfun 5588    Fn wfn 5589   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602
This theorem is referenced by:  setsidvald  14670
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