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Theorem resabs2 5138
Description: Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
Assertion
Ref Expression
resabs2  |-  ( B 
C_  C  ->  (
( A  |`  B )  |`  C )  =  ( A  |`  B )
)

Proof of Theorem resabs2
StepHypRef Expression
1 rescom 5132 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
2 resabs1 5136 . 2  |-  ( B 
C_  C  ->  (
( A  |`  C )  |`  B )  =  ( A  |`  B )
)
31, 2syl5eq 2499 1  |-  ( B 
C_  C  ->  (
( A  |`  B )  |`  C )  =  ( A  |`  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1446    C_ wss 3406    |` cres 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-opab 4465  df-xp 4843  df-rel 4844  df-res 4849
This theorem is referenced by:  residm  5139  fresaunres2  5760  fourierdlem104  38084  fouriersw  38105
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