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

Step	Hyp	Ref	Expression
1		rescom 5132	. 2 |- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )
2		resabs1 5136	. 2 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )
3	1, 2	syl5eq 2499	1 |- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) )

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