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Theorem resindm 4700
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 4697 . . 3  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
21ineq2d 3183 . 2  |-  ( Rel 
A  ->  ( ( A  |`  B )  i^i  ( A  |`  dom  A
) )  =  ( ( A  |`  B )  i^i  A ) )
3 resindi 4675 . 2  |-  ( A  |`  ( B  i^i  dom  A ) )  =  ( ( A  |`  B )  i^i  ( A  |`  dom  A ) )
4 incom 3174 . . 3  |-  ( ( A  |`  B )  i^i  A )  =  ( A  i^i  ( A  |`  B ) )
5 inres 4677 . . 3  |-  ( A  i^i  ( A  |`  B ) )  =  ( ( A  i^i  A )  |`  B )
6 inidm 3191 . . . 4  |-  ( A  i^i  A )  =  A
76reseq1i 4656 . . 3  |-  ( ( A  i^i  A )  |`  B )  =  ( A  |`  B )
84, 5, 73eqtrri 2108 . 2  |-  ( A  |`  B )  =  ( ( A  |`  B )  i^i  A )
92, 3, 83eqtr4g 2140 1  |-  ( Rel 
A  ->  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    i^i cin 2981   dom cdm 4391    |` cres 4393   Rel wrel 4396
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-dm 4401  df-res 4403
This theorem is referenced by:  resdmdfsn  4701
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