Theorem resabs1 3372

Description: Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25.

Assertion

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

Proof of Theorem resabs1

Step	Hyp	Ref	Expression
1		sseqin2 2219	. . 3 |- (B (_ C <-> (C i^i B) = B)
2		reseq2 3353	. . 3 |- ((C i^i B) = B -> (A |` (C i^i B)) = (A |` B))
3	1, 2	sylbi 199	. 2 |- (B (_ C -> (A |` (C i^i B)) = (A |` B))
4		resres 3361	. 2 |- ((A |` C) |` B) = (A |` (C i^i B))
5	3, 4	syl5eq 1511	1 |- (B (_ C -> ((A |` C) |` B) = (A |` B))

Syntax hints:   -> wi 3   = wceq 953   i^i cin 2036   (_ wss 2037   |` cres 3162

This theorem is referenced by:  resabs2 3373  resiima 3403  fun2ssres 3539  fssres2 3629  fvres 3719  tfrlem5 3900  dfrelog 8678  relogf1o 8679

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-opab 2657  df-xp 3174  df-rel 3175  df-res 3180

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