Theorem resabs2 4990

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 4984	. 2 |- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )
2		resabs1 4988	. 2 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )
3	1, 2	eqtrid 2250	1 |- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3166   |` cres 4677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4106  df-xp 4681  df-rel 4682  df-res 4687

This theorem is referenced by:  residm  4991