Theorem resdmdfsn 5106

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 5105	. 2 |- ( Rel R -> ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( _V \ { X } ) ) )
2		indif1 3476	. . . 4 |- ( ( _V \ { X } ) i^i dom R ) = ( ( _V i^i dom R ) \ { X } )
3		incom 3421	. . . . . 6 |- ( _V i^i dom R ) = ( dom R i^i _V )
4		inv1 3559	. . . . . 6 |- ( dom R i^i _V ) = dom R
5	3, 4	eqtri 2259	. . . . 5 |- ( _V i^i dom R ) = dom R
6	5	difeq1i 3343	. . . 4 |- ( ( _V i^i dom R ) \ { X } ) = ( dom R \ { X } )
7	2, 6	eqtri 2259	. . 3 |- ( ( _V \ { X } ) i^i dom R ) = ( dom R \ { X } )
8	7	reseq2i 5060	. 2 |- ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( dom R \ { X } ) )
9	1, 8	eqtr3di 2286	1 |- ( Rel R -> ( R |` ( _V \ { X } ) ) = ( R |` ( dom R \ { X } ) ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   _Vcvv 2821   \ cdif 3217   i^i cin 3219   {csn 3709   dom cdm 4774   |` cres 4776   Rel wrel 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-dm 4784  df-res 4786

This theorem is used by:  funresdfunsnss  5918  funresdfunsndc  6779