Theorem resdmdfsn 5002

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 5001	. 2 |- ( Rel R -> ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( _V \ { X } ) ) )
2		indif1 3418	. . . 4 |- ( ( _V \ { X } ) i^i dom R ) = ( ( _V i^i dom R ) \ { X } )
3		incom 3365	. . . . . 6 |- ( _V i^i dom R ) = ( dom R i^i _V )
4		inv1 3497	. . . . . 6 |- ( dom R i^i _V ) = dom R
5	3, 4	eqtri 2226	. . . . 5 |- ( _V i^i dom R ) = dom R
6	5	difeq1i 3287	. . . 4 |- ( ( _V i^i dom R ) \ { X } ) = ( dom R \ { X } )
7	2, 6	eqtri 2226	. . 3 |- ( ( _V \ { X } ) i^i dom R ) = ( dom R \ { X } )
8	7	reseq2i 4956	. 2 |- ( R |` ( ( _V \ { X } ) i^i dom R ) ) = ( R |` ( dom R \ { X } ) )
9	1, 8	eqtr3di 2253	1 |- ( Rel R -> ( R |` ( _V \ { X } ) ) = ( R |` ( dom R \ { X } ) ) )

Syntax hints:   -> wi 4   = wceq 1373   _Vcvv 2772   \ cdif 3163   i^i cin 3165   {csn 3633   dom cdm 4675   |` cres 4677   Rel wrel 4680

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-dm 4685  df-res 4687

This theorem is referenced by:  funresdfunsnss  5787  funresdfunsndc  6592