Theorem resabs2i 43815

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 6012	. 2 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1542   ⊆ wss 3948   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-xp 5682  df-rel 5683  df-res 5688

This theorem is referenced by: (None)