Theorem resindm 4933

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 4930	. . 3 |- ( Rel A -> ( A |` dom A ) = A )
2	1	ineq2d 3328	. 2 |- ( Rel A -> ( ( A |` B ) i^i ( A |` dom A ) ) = ( ( A |` B ) i^i A ) )
3		resindi 4906	. 2 |- ( A |` ( B i^i dom A ) ) = ( ( A |` B ) i^i ( A |` dom A ) )
4		incom 3319	. . 3 |- ( ( A |` B ) i^i A ) = ( A i^i ( A |` B ) )
5		inres 4908	. . 3 |- ( A i^i ( A |` B ) ) = ( ( A i^i A ) |` B )
6		inidm 3336	. . . 4 |- ( A i^i A ) = A
7	6	reseq1i 4887	. . 3 |- ( ( A i^i A ) |` B ) = ( A |` B )
8	4, 5, 7	3eqtrri 2196	. 2 |- ( A |` B ) = ( ( A |` B ) i^i A )
9	2, 3, 8	3eqtr4g 2228	1 |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1348   i^i cin 3120   dom cdm 4611   |` cres 4613   Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-dm 4621  df-res 4623

This theorem is referenced by:  resdmdfsn  4934