Theorem resabs2 5245

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 5239	. 2 |- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )
2		resabs1 5243	. 2 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )
3	1, 2	syl5eq 2455	1 |- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1405   C_ wss 3413   |` cres 4944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-opab 4453  df-xp 4948  df-rel 4949  df-res 4954

This theorem is referenced by:  residm  5246  fresaunres2  5696  fourierdlem104  37343  fouriersw  37364