MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resdmdfsn Structured version   Unicode version

Theorem resdmdfsn 5307
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 5306 . 2  |-  ( Rel 
R  ->  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( _V  \  { X } ) ) )
2 indif1 3739 . . . . 5  |-  ( ( _V  \  { X } )  i^i  dom  R )  =  ( ( _V  i^i  dom  R
)  \  { X } )
3 incom 3677 . . . . . . 7  |-  ( _V 
i^i  dom  R )  =  ( dom  R  i^i  _V )
4 inv1 3811 . . . . . . 7  |-  ( dom 
R  i^i  _V )  =  dom  R
53, 4eqtri 2483 . . . . . 6  |-  ( _V 
i^i  dom  R )  =  dom  R
65difeq1i 3604 . . . . 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 5262 . 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 1398   _Vcvv 3106    \ cdif 3458    i^i cin 3460   {csn 4016   dom cdm 4988    |` cres 4990   Rel wrel 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-dm 4998  df-res 5000
This theorem is referenced by:  funresdfunsn  6089
  Copyright terms: Public domain W3C validator