Theorem resdmdfsn 4862

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

Step	Hyp	Ref	Expression
1		indif1 3321	. . . 4 |- ( ( _V \ { X } ) i^i dom R ) = ( ( _V i^i dom R ) \ { X } )
2		incom 3268	. . . . . 6 |- ( _V i^i dom R ) = ( dom R i^i _V )
3		inv1 3399	. . . . . 6 |- ( dom R i^i _V ) = dom R
4	2, 3	eqtri 2160	. . . . 5 |- ( _V i^i dom R ) = dom R
5	4	difeq1i 3190	. . . 4 |- ( ( _V i^i dom R ) \ { X } ) = ( dom R \ { X } )
6	1, 5	eqtri 2160	. . 3 |- ( ( _V \ { X } ) i^i dom R ) = ( dom R \ { X } )
7	6	reseq2i 4816	. 2 |- ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( dom R \ { X } ) )
8		resindm 4861	. 2 |- ( Rel R -> ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( _V \ { X } ) ) )
9	7, 8	syl5reqr 2187	1 |- ( Rel R -> ( R |` ( _V \ { X } ) ) = ( R |` ( dom R \ { X } ) ) )

Syntax hints:   -> wi 4   = wceq 1331   _Vcvv 2686   \ cdif 3068   i^i cin 3070   {csn 3527   dom cdm 4539   |` cres 4541   Rel wrel 4544

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-dm 4549  df-res 4551

This theorem is referenced by:  funresdfunsnss  5623  funresdfunsndc  6402