Theorem resabs1 4658

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 4642	. 2 |- ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
2		sseqin2 3185	. . 3 |- ( B C_ C <-> ( C i^i B ) = B )
3		reseq2 4625	. . 3 |- ( ( C i^i B ) = B -> ( A |` ( C i^i B ) ) = ( A |` B ) )
4	2, 3	sylbi 119	. 2 |- ( B C_ C -> ( A |` ( C i^i B ) ) = ( A |` B ) )
5	1, 4	syl5eq 2125	1 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1284   i^i cin 2972   C_ wss 2973   |` cres 4365

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-xp 4369  df-rel 4370  df-res 4375

This theorem is referenced by:  resabs2  4659  resiima  4703  fun2ssres  4963  fssres2  5087  f2ndf  5867  smores3  5931  tfrlemisucaccv  5962