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Theorem resdmdfsn 4962
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 4961 . 2  |-  ( Rel 
R  ->  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( _V  \  { X } ) ) )
2 indif1 3392 . . . 4  |-  ( ( _V  \  { X } )  i^i  dom  R )  =  ( ( _V  i^i  dom  R
)  \  { X } )
3 incom 3339 . . . . . 6  |-  ( _V 
i^i  dom  R )  =  ( dom  R  i^i  _V )
4 inv1 3471 . . . . . 6  |-  ( dom 
R  i^i  _V )  =  dom  R
53, 4eqtri 2208 . . . . 5  |-  ( _V 
i^i  dom  R )  =  dom  R
65difeq1i 3261 . . . 4  |-  ( ( _V  i^i  dom  R
)  \  { X } )  =  ( dom  R  \  { X } )
72, 6eqtri 2208 . . 3  |-  ( ( _V  \  { X } )  i^i  dom  R )  =  ( dom 
R  \  { X } )
87reseq2i 4916 . 2  |-  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( dom  R  \  { X } ) )
91, 8eqtr3di 2235 1  |-  ( Rel 
R  ->  ( R  |`  ( _V  \  { X } ) )  =  ( R  |`  ( dom  R  \  { X } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363   _Vcvv 2749    \ cdif 3138    i^i cin 3140   {csn 3604   dom cdm 4638    |` cres 4640   Rel wrel 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-dm 4648  df-res 4650
This theorem is referenced by:  funresdfunsnss  5732  funresdfunsndc  6521
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