Theorem resabs1 4640

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 4624	. 2 |- ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
2		sseqin2 3156	. . 3 |- ( B C_ C <-> ( C i^i B ) = B )
3		reseq2 4607	. . 3 |- ( ( C i^i B ) = B -> ( A |` ( C i^i B ) ) = ( A |` B ) )
4	2, 3	sylbi 114	. 2 |- ( B C_ C -> ( A |` ( C i^i B ) ) = ( A |` B ) )
5	1, 4	syl5eq 2084	1 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1243   i^i cin 2916   C_ wss 2917   |` cres 4347

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352  df-res 4357

This theorem is referenced by:  resabs2  4641  resiima  4683  fun2ssres  4943  fssres2  5067  f2ndf  5847  smores3  5908  tfrlemisucaccv  5939