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Theorem resdmdfsn 4788
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 indif1 3260 . . . 4  |-  ( ( _V  \  { X } )  i^i  dom  R )  =  ( ( _V  i^i  dom  R
)  \  { X } )
2 incom 3207 . . . . . 6  |-  ( _V 
i^i  dom  R )  =  ( dom  R  i^i  _V )
3 inv1 3338 . . . . . 6  |-  ( dom 
R  i^i  _V )  =  dom  R
42, 3eqtri 2115 . . . . 5  |-  ( _V 
i^i  dom  R )  =  dom  R
54difeq1i 3129 . . . 4  |-  ( ( _V  i^i  dom  R
)  \  { X } )  =  ( dom  R  \  { X } )
61, 5eqtri 2115 . . 3  |-  ( ( _V  \  { X } )  i^i  dom  R )  =  ( dom 
R  \  { X } )
76reseq2i 4742 . 2  |-  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( dom  R  \  { X } ) )
8 resindm 4787 . 2  |-  ( Rel 
R  ->  ( R  |`  ( ( _V  \  { X } )  i^i 
dom  R ) )  =  ( R  |`  ( _V  \  { X } ) ) )
97, 8syl5reqr 2142 1  |-  ( Rel 
R  ->  ( R  |`  ( _V  \  { X } ) )  =  ( R  |`  ( dom  R  \  { X } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1296   _Vcvv 2633    \ cdif 3010    i^i cin 3012   {csn 3466   dom cdm 4467    |` cres 4469   Rel wrel 4472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-xp 4473  df-rel 4474  df-dm 4477  df-res 4479
This theorem is referenced by:  funresdfunsnss  5539  funresdfunsndc  6305
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