Theorem resdmdfsn 4932

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 4931	. 2 |- ( Rel R -> ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( _V \ { X } ) ) )
2		indif1 3372	. . . 4 |- ( ( _V \ { X } ) i^i dom R ) = ( ( _V i^i dom R ) \ { X } )
3		incom 3319	. . . . . 6 |- ( _V i^i dom R ) = ( dom R i^i _V )
4		inv1 3450	. . . . . 6 |- ( dom R i^i _V ) = dom R
5	3, 4	eqtri 2191	. . . . 5 |- ( _V i^i dom R ) = dom R
6	5	difeq1i 3241	. . . 4 |- ( ( _V i^i dom R ) \ { X } ) = ( dom R \ { X } )
7	2, 6	eqtri 2191	. . 3 |- ( ( _V \ { X } ) i^i dom R ) = ( dom R \ { X } )
8	7	reseq2i 4886	. 2 |- ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( dom R \ { X } ) )
9	1, 8	eqtr3di 2218	1 |- ( Rel R -> ( R |` ( _V \ { X } ) ) = ( R |` ( dom R \ { X } ) ) )

Syntax hints:   -> wi 4   = wceq 1348   _Vcvv 2730   \ cdif 3118   i^i cin 3120   {csn 3581   dom cdm 4609   |` cres 4611   Rel wrel 4614

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-dm 4619  df-res 4621

This theorem is referenced by:  funresdfunsnss  5697  funresdfunsndc  6483