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Theorem resindm 5151
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 5126 . 2  |-  ( A  |`  ( B  i^i  dom  A ) )  =  ( ( A  |`  B )  i^i  ( A  |`  dom  A ) )
2 resdm 5148 . . . 4  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
32ineq2d 3552 . . 3  |-  ( Rel 
A  ->  ( ( A  |`  B )  i^i  ( A  |`  dom  A
) )  =  ( ( A  |`  B )  i^i  A ) )
4 incom 3543 . . . . 5  |-  ( ( A  |`  B )  i^i  A )  =  ( A  i^i  ( A  |`  B ) )
5 inres 5128 . . . . 5  |-  ( A  i^i  ( A  |`  B ) )  =  ( ( A  i^i  A )  |`  B )
64, 5eqtri 2463 . . . 4  |-  ( ( A  |`  B )  i^i  A )  =  ( ( A  i^i  A
)  |`  B )
7 inidm 3559 . . . . . 6  |-  ( A  i^i  A )  =  A
87reseq1i 5106 . . . . 5  |-  ( ( A  i^i  A )  |`  B )  =  ( A  |`  B )
98a1i 11 . . . 4  |-  ( Rel 
A  ->  ( ( A  i^i  A )  |`  B )  =  ( A  |`  B )
)
106, 9syl5eq 2487 . . 3  |-  ( Rel 
A  ->  ( ( A  |`  B )  i^i 
A )  =  ( A  |`  B )
)
113, 10eqtrd 2475 . 2  |-  ( Rel 
A  ->  ( ( A  |`  B )  i^i  ( A  |`  dom  A
) )  =  ( A  |`  B )
)
121, 11syl5eq 2487 1  |-  ( Rel 
A  ->  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    i^i cin 3327   dom cdm 4840    |` cres 4842   Rel wrel 4845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-rel 4847  df-dm 4850  df-res 4852
This theorem is referenced by:  resdmdfsn  5152
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