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Theorem resdmdfsn 5305
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 5304 . 2  |-  ( Rel 
R  ->  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( _V  \  { X } ) ) )
2 indif1 3724 . . . . 5  |-  ( ( _V  \  { X } )  i^i  dom  R )  =  ( ( _V  i^i  dom  R
)  \  { X } )
3 incom 3673 . . . . . . 7  |-  ( _V 
i^i  dom  R )  =  ( dom  R  i^i  _V )
4 inv1 3794 . . . . . . 7  |-  ( dom 
R  i^i  _V )  =  dom  R
53, 4eqtri 2470 . . . . . 6  |-  ( _V 
i^i  dom  R )  =  dom  R
65difeq1i 3600 . . . . 5  |-  ( ( _V  i^i  dom  R
)  \  { X } )  =  ( dom  R  \  { X } )
72, 6eqtri 2470 . . . 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 5259 . 2  |-  ( Rel 
R  ->  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( dom  R  \  { X } ) ) )
101, 9eqtr3d 2484 1  |-  ( Rel 
R  ->  ( R  |`  ( _V  \  { X } ) )  =  ( R  |`  ( dom  R  \  { X } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381   _Vcvv 3093    \ cdif 3455    i^i cin 3457   {csn 4010   dom cdm 4985    |` cres 4987   Rel wrel 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-xp 4991  df-rel 4992  df-dm 4995  df-res 4997
This theorem is referenced by:  funresdfunsn  6094
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