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

Step	Hyp	Ref	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
4	2, 3	eqtri 2115	. . . . 5 |- ( _V i^i dom R ) = dom R
5	4	difeq1i 3129	. . . 4 |- ( ( _V i^i dom R ) \ { X } ) = ( dom R \ { X } )
6	1, 5	eqtri 2115	. . 3 |- ( ( _V \ { X } ) i^i dom R ) = ( dom R \ { X } )
7	6	reseq2i 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 } ) ) )
9	7, 8	syl5reqr 2142	1 |- ( Rel R -> ( R |` ( _V \ { X } ) ) = ( R |` ( dom R \ { X } ) ) )

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