Theorem resindm 4670

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 4667	. . 3 |- ( Rel A -> ( A |` dom A ) = A )
2	1	ineq2d 3167	. 2 |- ( Rel A -> ( ( A |` B ) i^i ( A |` dom A ) ) = ( ( A |` B ) i^i A ) )
3		resindi 4645	. 2 |- ( A |` ( B i^i dom A ) ) = ( ( A |` B ) i^i ( A |` dom A ) )
4		incom 3158	. . 3 |- ( ( A |` B ) i^i A ) = ( A i^i ( A |` B ) )
5		inres 4647	. . 3 |- ( A i^i ( A |` B ) ) = ( ( A i^i A ) |` B )
6		inidm 3175	. . . 4 |- ( A i^i A ) = A
7	6	reseq1i 4626	. . 3 |- ( ( A i^i A ) |` B ) = ( A |` B )
8	4, 5, 7	3eqtrri 2106	. 2 |- ( A |` B ) = ( ( A |` B ) i^i A )
9	2, 3, 8	3eqtr4g 2138	1 |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1284   i^i cin 2972   dom cdm 4363   |` cres 4365   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-dm 4373  df-res 4375

This theorem is referenced by:  resdmdfsn  4671