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Theorem resdmdfsn 5147
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
resdmdfsn  |-  ( Rel 
R  ->  ( R  |`  ( _V  \  { X } ) )  =  ( R  |`  ( dom  R  \  { X } ) ) )

Proof of Theorem resdmdfsn
StepHypRef Expression
1 resindm 5146 . 2  |-  ( Rel 
R  ->  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( _V  \  { X } ) ) )
2 indif1 3589 . . . . 5  |-  ( ( _V  \  { X } )  i^i  dom  R )  =  ( ( _V  i^i  dom  R
)  \  { X } )
3 incom 3538 . . . . . . 7  |-  ( _V 
i^i  dom  R )  =  ( dom  R  i^i  _V )
4 inv1 3659 . . . . . . 7  |-  ( dom 
R  i^i  _V )  =  dom  R
53, 4eqtri 2458 . . . . . 6  |-  ( _V 
i^i  dom  R )  =  dom  R
65difeq1i 3465 . . . . 5  |-  ( ( _V  i^i  dom  R
)  \  { X } )  =  ( dom  R  \  { X } )
72, 6eqtri 2458 . . . 4  |-  ( ( _V  \  { X } )  i^i  dom  R )  =  ( dom 
R  \  { X } )
87a1i 11 . . 3  |-  ( Rel 
R  ->  ( ( _V  \  { X }
)  i^i  dom  R )  =  ( dom  R  \  { X } ) )
98reseq2d 5105 . 2  |-  ( Rel 
R  ->  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( dom  R  \  { X } ) ) )
101, 9eqtr3d 2472 1  |-  ( Rel 
R  ->  ( R  |`  ( _V  \  { X } ) )  =  ( R  |`  ( dom  R  \  { X } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   _Vcvv 2967    \ cdif 3320    i^i cin 3322   {csn 3872   dom cdm 4835    |` cres 4837   Rel wrel 4840
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 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-dm 4845  df-res 4847
This theorem is referenced by:  funresdfunsn  5915
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