Theorem resindm 5100

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 5097	. . 3 |- ( Rel A -> ( A |` dom A ) = A )
2	1	ineq2d 3432	. 2 |- ( Rel A -> ( ( A |` B ) i^i ( A |` dom A ) ) = ( ( A |` B ) i^i A ) )
3		resindi 5073	. 2 |- ( A |` ( B i^i dom A ) ) = ( ( A |` B ) i^i ( A |` dom A ) )
4		incom 3421	. . 3 |- ( ( A |` B ) i^i A ) = ( A i^i ( A |` B ) )
5		inres 5075	. . 3 |- ( A i^i ( A |` B ) ) = ( ( A i^i A ) |` B )
6		inidm 3440	. . . 4 |- ( A i^i A ) = A
7	6	reseq1i 5054	. . 3 |- ( ( A i^i A ) |` B ) = ( A |` B )
8	4, 5, 7	3eqtrri 2264	. 2 |- ( A |` B ) = ( ( A |` B ) i^i A )
9	2, 3, 8	3eqtr4g 2296	1 |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1402   i^i cin 3219   dom cdm 4769   |` cres 4771   Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-dm 4779  df-res 4781

This theorem is referenced by:  resdmdfsn  5101