Theorem resdmdfsn 5021

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		resindm 5020	. 2 |- ( Rel R -> ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( _V \ { X } ) ) )
2		indif1 3426	. . . 4 |- ( ( _V \ { X } ) i^i dom R ) = ( ( _V i^i dom R ) \ { X } )
3		incom 3373	. . . . . 6 |- ( _V i^i dom R ) = ( dom R i^i _V )
4		inv1 3505	. . . . . 6 |- ( dom R i^i _V ) = dom R
5	3, 4	eqtri 2228	. . . . 5 |- ( _V i^i dom R ) = dom R
6	5	difeq1i 3295	. . . 4 |- ( ( _V i^i dom R ) \ { X } ) = ( dom R \ { X } )
7	2, 6	eqtri 2228	. . 3 |- ( ( _V \ { X } ) i^i dom R ) = ( dom R \ { X } )
8	7	reseq2i 4975	. 2 |- ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( dom R \ { X } ) )
9	1, 8	eqtr3di 2255	1 |- ( Rel R -> ( R |` ( _V \ { X } ) ) = ( R |` ( dom R \ { X } ) ) )

Syntax hints:   -> wi 4   = wceq 1373   _Vcvv 2776   \ cdif 3171   i^i cin 3173   {csn 3643   dom cdm 4693   |` cres 4695   Rel wrel 4698

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-dm 4703  df-res 4705

This theorem is referenced by:  funresdfunsnss  5810  funresdfunsndc  6615