Theorem resdmdfsn 4927

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 4926	. 2 |- ( Rel R -> ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( _V \ { X } ) ) )
2		indif1 3367	. . . 4 |- ( ( _V \ { X } ) i^i dom R ) = ( ( _V i^i dom R ) \ { X } )
3		incom 3314	. . . . . 6 |- ( _V i^i dom R ) = ( dom R i^i _V )
4		inv1 3445	. . . . . 6 |- ( dom R i^i _V ) = dom R
5	3, 4	eqtri 2186	. . . . 5 |- ( _V i^i dom R ) = dom R
6	5	difeq1i 3236	. . . 4 |- ( ( _V i^i dom R ) \ { X } ) = ( dom R \ { X } )
7	2, 6	eqtri 2186	. . 3 |- ( ( _V \ { X } ) i^i dom R ) = ( dom R \ { X } )
8	7	reseq2i 4881	. 2 |- ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( dom R \ { X } ) )
9	1, 8	eqtr3di 2214	1 |- ( Rel R -> ( R |` ( _V \ { X } ) ) = ( R |` ( dom R \ { X } ) ) )

Syntax hints:   -> wi 4   = wceq 1343   _Vcvv 2726   \ cdif 3113   i^i cin 3115   {csn 3576   dom cdm 4604   |` cres 4606   Rel wrel 4609

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-dm 4614  df-res 4616

This theorem is referenced by:  funresdfunsnss  5688  funresdfunsndc  6474