Theorem residm 6011

Description: Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)

Assertion

Ref	Expression
residm	⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵)

Proof of Theorem residm

Step	Hyp	Ref	Expression
1		ssid 3960	. 2 ⊢ 𝐵 ⊆ 𝐵
2		resabs2 6010	. 2 ⊢ (𝐵 ⊆ 𝐵 → ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵)

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3906   ↾ cres 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5176  df-xp 5669  df-rel 5670  df-res 5675

This theorem is used by:  resima  6016  dffv2  6980  fvsnun2  7187  qtopres  23908  bnj1253  35472  eldioph2lem1  43551  eldioph2lem2  43552  relexpiidm  44490