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Theorem resdmdfsn 5259
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 5258 . 2  |-  ( Rel 
R  ->  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( _V  \  { X } ) ) )
2 indif1 3701 . . . . 5  |-  ( ( _V  \  { X } )  i^i  dom  R )  =  ( ( _V  i^i  dom  R
)  \  { X } )
3 incom 3650 . . . . . . 7  |-  ( _V 
i^i  dom  R )  =  ( dom  R  i^i  _V )
4 inv1 3771 . . . . . . 7  |-  ( dom 
R  i^i  _V )  =  dom  R
53, 4eqtri 2483 . . . . . 6  |-  ( _V 
i^i  dom  R )  =  dom  R
65difeq1i 3577 . . . . 5  |-  ( ( _V  i^i  dom  R
)  \  { X } )  =  ( dom  R  \  { X } )
72, 6eqtri 2483 . . . 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 5217 . 2  |-  ( Rel 
R  ->  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( dom  R  \  { X } ) ) )
101, 9eqtr3d 2497 1  |-  ( Rel 
R  ->  ( R  |`  ( _V  \  { X } ) )  =  ( R  |`  ( dom  R  \  { X } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   _Vcvv 3076    \ cdif 3432    i^i cin 3434   {csn 3984   dom cdm 4947    |` cres 4949   Rel wrel 4952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-xp 4953  df-rel 4954  df-dm 4957  df-res 4959
This theorem is referenced by:  funresdfunsn  6028
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