Theorem resindm 4951

Description: When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.)

Assertion

Ref	Expression
resindm	|- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )

Proof of Theorem resindm

Step	Hyp	Ref	Expression
1		resdm 4948	. . 3 |- ( Rel A -> ( A |` dom A ) = A )
2	1	ineq2d 3338	. 2 |- ( Rel A -> ( ( A |` B ) i^i ( A |` dom A ) ) = ( ( A |` B ) i^i A ) )
3		resindi 4924	. 2 |- ( A |` ( B i^i dom A ) ) = ( ( A |` B ) i^i ( A |` dom A ) )
4		incom 3329	. . 3 |- ( ( A |` B ) i^i A ) = ( A i^i ( A |` B ) )
5		inres 4926	. . 3 |- ( A i^i ( A |` B ) ) = ( ( A i^i A ) |` B )
6		inidm 3346	. . . 4 |- ( A i^i A ) = A
7	6	reseq1i 4905	. . 3 |- ( ( A i^i A ) |` B ) = ( A |` B )
8	4, 5, 7	3eqtrri 2203	. 2 |- ( A |` B ) = ( ( A |` B ) i^i A )
9	2, 3, 8	3eqtr4g 2235	1 |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1353   i^i cin 3130   dom cdm 4628   |` cres 4630   Rel wrel 4633

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-dm 4638  df-res 4640

This theorem is referenced by:  resdmdfsn  4952