Theorem resindm 4942

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 4939	. . 3 |- ( Rel A -> ( A |` dom A ) = A )
2	1	ineq2d 3334	. 2 |- ( Rel A -> ( ( A |` B ) i^i ( A |` dom A ) ) = ( ( A |` B ) i^i A ) )
3		resindi 4915	. 2 |- ( A |` ( B i^i dom A ) ) = ( ( A |` B ) i^i ( A |` dom A ) )
4		incom 3325	. . 3 |- ( ( A |` B ) i^i A ) = ( A i^i ( A |` B ) )
5		inres 4917	. . 3 |- ( A i^i ( A |` B ) ) = ( ( A i^i A ) |` B )
6		inidm 3342	. . . 4 |- ( A i^i A ) = A
7	6	reseq1i 4896	. . 3 |- ( ( A i^i A ) |` B ) = ( A |` B )
8	4, 5, 7	3eqtrri 2201	. 2 |- ( A |` B ) = ( ( A |` B ) i^i A )
9	2, 3, 8	3eqtr4g 2233	1 |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1353   i^i cin 3126   dom cdm 4620   |` cres 4622   Rel wrel 4625

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-dm 4630  df-res 4632

This theorem is referenced by:  resdmdfsn  4943