Theorem resabs1 4948

Description: Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)

Assertion

Ref	Expression
resabs1	|- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )

Proof of Theorem resabs1

Step	Hyp	Ref	Expression
1		resres 4931	. 2 |- ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
2		sseqin2 3366	. . 3 |- ( B C_ C <-> ( C i^i B ) = B )
3		reseq2 4914	. . 3 |- ( ( C i^i B ) = B -> ( A |` ( C i^i B ) ) = ( A |` B ) )
4	2, 3	sylbi 121	. 2 |- ( B C_ C -> ( A |` ( C i^i B ) ) = ( A |` B ) )
5	1, 4	eqtrid 2232	1 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1363   i^i cin 3140   C_ wss 3141   |` cres 4640

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-opab 4077  df-xp 4644  df-rel 4645  df-res 4650

This theorem is referenced by:  resabs1d  4949  resabs2  4950  resiima  4998  fun2ssres  5271  fssres2  5405  f2ndf  6241  smores3  6308  tfrlemisucaccv  6340  tfr1onlemsucaccv  6356  tfrcllemsucaccv  6369  djudom  7106  setsresg  12514