Theorem resabs2d 44100

Description: Absorption law for restriction. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
resabs2d.1	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
resabs2d	⊢ (𝜑 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵))

Proof of Theorem resabs2d

Step	Hyp	Ref	Expression
1		resabs2d.1	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
2		resabs2 6011	. 2 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1541   ⊆ wss 3947   ↾ cres 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-opab 5210  df-xp 5681  df-rel 5682  df-res 5687

This theorem is referenced by: (None)