Theorem resabs1 3394

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 2232	. . 3 ⊢ (B ⊆ C ↔ (C ∩ B) = B)
2		reseq2 3375	. . 3 ⊢ ((C ∩ B) = B → (A ↾ (C ∩ B)) = (A ↾ B))
3	1, 2	sylbi 199	. 2 ⊢ (B ⊆ C → (A ↾ (C ∩ B)) = (A ↾ B))
4		resres 3383	. 2 ⊢ ((A ↾ C) ↾ B) = (A ↾ (C ∩ B))
5	3, 4	syl5eq 1522	1 ⊢ (B ⊆ C → ((A ↾ C) ↾ B) = (A ↾ B))

Syntax hints:   → wi 3   = wceq 958   ∩ cin 2049   ⊆ wss 2050   ↾ cres 3178

This theorem is referenced by:  resabs2 3395  resiima 3425  fun2ssres 3559  fssres2 3650  fvres 3740  tfrlem5 3921  dfrelog 8751  relogf1o 8752

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-xp 3190  df-rel 3191  df-res 3196

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