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Theorem resdmdfsn 4932
Description: Restricting a relation to its domain without a set is the same as restricting the relation to the universe without this set. (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 4931 . 2  |-  ( Rel 
R  ->  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( _V  \  { X } ) ) )
2 indif1 3372 . . . 4  |-  ( ( _V  \  { X } )  i^i  dom  R )  =  ( ( _V  i^i  dom  R
)  \  { X } )
3 incom 3319 . . . . . 6  |-  ( _V 
i^i  dom  R )  =  ( dom  R  i^i  _V )
4 inv1 3450 . . . . . 6  |-  ( dom 
R  i^i  _V )  =  dom  R
53, 4eqtri 2191 . . . . 5  |-  ( _V 
i^i  dom  R )  =  dom  R
65difeq1i 3241 . . . 4  |-  ( ( _V  i^i  dom  R
)  \  { X } )  =  ( dom  R  \  { X } )
72, 6eqtri 2191 . . 3  |-  ( ( _V  \  { X } )  i^i  dom  R )  =  ( dom 
R  \  { X } )
87reseq2i 4886 . 2  |-  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( dom  R  \  { X } ) )
91, 8eqtr3di 2218 1  |-  ( Rel 
R  ->  ( R  |`  ( _V  \  { X } ) )  =  ( R  |`  ( dom  R  \  { X } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348   _Vcvv 2730    \ cdif 3118    i^i cin 3120   {csn 3581   dom cdm 4609    |` cres 4611   Rel wrel 4614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-dm 4619  df-res 4621
This theorem is referenced by:  funresdfunsnss  5697  funresdfunsndc  6483
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