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Theorem resindm 5324
Description: Class restriction distributes over intersection. (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 resindi 5295 . 2  |-  ( A  |`  ( B  i^i  dom  A ) )  =  ( ( A  |`  B )  i^i  ( A  |`  dom  A ) )
2 resdm 5321 . . . 4  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
32ineq2d 3705 . . 3  |-  ( Rel 
A  ->  ( ( A  |`  B )  i^i  ( A  |`  dom  A
) )  =  ( ( A  |`  B )  i^i  A ) )
4 incom 3696 . . . . 5  |-  ( ( A  |`  B )  i^i  A )  =  ( A  i^i  ( A  |`  B ) )
5 inres 5297 . . . . 5  |-  ( A  i^i  ( A  |`  B ) )  =  ( ( A  i^i  A )  |`  B )
64, 5eqtri 2496 . . . 4  |-  ( ( A  |`  B )  i^i  A )  =  ( ( A  i^i  A
)  |`  B )
7 inidm 3712 . . . . . 6  |-  ( A  i^i  A )  =  A
87reseq1i 5275 . . . . 5  |-  ( ( A  i^i  A )  |`  B )  =  ( A  |`  B )
98a1i 11 . . . 4  |-  ( Rel 
A  ->  ( ( A  i^i  A )  |`  B )  =  ( A  |`  B )
)
106, 9syl5eq 2520 . . 3  |-  ( Rel 
A  ->  ( ( A  |`  B )  i^i 
A )  =  ( A  |`  B )
)
113, 10eqtrd 2508 . 2  |-  ( Rel 
A  ->  ( ( A  |`  B )  i^i  ( A  |`  dom  A
) )  =  ( A  |`  B )
)
121, 11syl5eq 2520 1  |-  ( Rel 
A  ->  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    i^i cin 3480   dom cdm 5005    |` cres 5007   Rel wrel 5010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-dm 5015  df-res 5017
This theorem is referenced by:  resdmdfsn  5325  resfifsupp  7869  fresin2  31349  cncfuni  31548  fourierdlem48  31778  fourierdlem49  31779  fourierdlem113  31843
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