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Theorem resindm 4831
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
StepHypRef Expression
1 resdm 4828 . . 3  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
21ineq2d 3247 . 2  |-  ( Rel 
A  ->  ( ( A  |`  B )  i^i  ( A  |`  dom  A
) )  =  ( ( A  |`  B )  i^i  A ) )
3 resindi 4804 . 2  |-  ( A  |`  ( B  i^i  dom  A ) )  =  ( ( A  |`  B )  i^i  ( A  |`  dom  A ) )
4 incom 3238 . . 3  |-  ( ( A  |`  B )  i^i  A )  =  ( A  i^i  ( A  |`  B ) )
5 inres 4806 . . 3  |-  ( A  i^i  ( A  |`  B ) )  =  ( ( A  i^i  A )  |`  B )
6 inidm 3255 . . . 4  |-  ( A  i^i  A )  =  A
76reseq1i 4785 . . 3  |-  ( ( A  i^i  A )  |`  B )  =  ( A  |`  B )
84, 5, 73eqtrri 2143 . 2  |-  ( A  |`  B )  =  ( ( A  |`  B )  i^i  A )
92, 3, 83eqtr4g 2175 1  |-  ( Rel 
A  ->  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    i^i cin 3040   dom cdm 4509    |` cres 4511   Rel wrel 4514
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-dm 4519  df-res 4521
This theorem is referenced by:  resdmdfsn  4832
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