Theorem resabs2 4914

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 4908	. 2 |- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )
2		resabs1 4912	. 2 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )
3	1, 2	syl5eq 2210	1 |- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1343   C_ wss 3115   |` cres 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-opab 4043  df-xp 4609  df-rel 4610  df-res 4615

This theorem is referenced by:  residm  4915