Theorem residm 5996

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 3958	. 2 ⊢ 𝐵 ⊆ 𝐵
2		resabs2 5995	. 2 ⊢ (𝐵 ⊆ 𝐵 → ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1560   ⊆ wss 3904   ↾ cres 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5653  df-rel 5654  df-res 5659

This theorem is referenced by:  resima  6001  dffv2  6962  fvsnun2  7167  qtopres  23758  bnj1253  35312  eldioph2lem1  43341  eldioph2lem2  43342  relexpiidm  44280