Theorem resabs2i 45131

Description: Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
resabs2i.1	⊢ 𝐵 ⊆ 𝐶

Assertion

Ref	Expression
resabs2i	⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵)

Proof of Theorem resabs2i

Step	Hyp	Ref	Expression
1		resabs2i.1	. 2 ⊢ 𝐵 ⊆ 𝐶
2		resabs2 6001	. 2 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1540   ⊆ wss 3931   ↾ cres 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-opab 5187  df-xp 5665  df-rel 5666  df-res 5671

This theorem is referenced by: (None)