Theorem resabs1 4976

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 4959	. 2 |- ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
2		sseqin2 3383	. . 3 |- ( B C_ C <-> ( C i^i B ) = B )
3		reseq2 4942	. . 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 2241	1 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1364   i^i cin 3156   C_ wss 3157   |` cres 4666

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-opab 4096  df-xp 4670  df-rel 4671  df-res 4676

This theorem is referenced by:  resabs1d  4977  resabs2  4978  resiima  5028  fun2ssres  5302  fssres2  5438  f2ndf  6293  smores3  6360  tfrlemisucaccv  6392  tfr1onlemsucaccv  6408  tfrcllemsucaccv  6421  djudom  7168  setsresg  12741