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Theorem resabs2 5145
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 5140 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
2 resabs1 5144 . 2  |-  ( B 
C_  C  ->  (
( A  |`  C )  |`  B )  =  ( A  |`  B )
)
31, 2syl5eq 2487 1  |-  ( B 
C_  C  ->  (
( A  |`  B )  |`  C )  =  ( A  |`  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    C_ wss 3333    |` cres 4847
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 4418  ax-nul 4426  ax-pr 4536
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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-opab 4356  df-xp 4851  df-rel 4852  df-res 4857
This theorem is referenced by:  residm  5146  fresaunres2  5588
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