Theorem resabs2 5429

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 5423	. 2 |- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )
2		resabs1 5427	. 2 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )
3	1, 2	syl5eq 2668	1 |- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1483   C_ wss 3574   |` cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121  df-res 5126

This theorem is referenced by:  residm  5430  fresaunres2  6076  resabs2i  39330  resabs2d  39629  fourierdlem104  40427  fouriersw  40448