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Theorem resabs2 5169
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 5164 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
2 resabs1 5168 . 2  |-  ( B 
C_  C  ->  (
( A  |`  C )  |`  B )  =  ( A  |`  B )
)
31, 2syl5eq 2480 1  |-  ( B 
C_  C  ->  (
( A  |`  B )  |`  C )  =  ( A  |`  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    C_ wss 3313    |` cres 4873
This theorem is referenced by:  residm  5170  fresaunres2  5608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-opab 4260  df-xp 4877  df-rel 4878  df-res 4883
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