Theorem resabs1 3388

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 2229	. . 3 |- (B (_ C <-> (C i^i B) = B)
2		reseq2 3369	. . 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 3377	. 2 |- ((A |` C) |` B) = (A |` (C i^i B))
5	3, 4	syl5eq 1519	1 |- (B (_ C -> ((A |` C) |` B) = (A |` B))

Syntax hints:   -> wi 3   = wceq 956   i^i cin 2046   (_ wss 2047   |` cres 3172

This theorem is referenced by:  resabs2 3389  resiima 3419  fun2ssres 3553  fssres2 3644  fvres 3734  tfrlem5 3915  dfrelog 8756  relogf1o 8757

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-xp 3184  df-rel 3185  df-res 3190

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