Theorem resabs2i 42659

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 5922	. 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 3892   ↾ cres 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5142  df-xp 5596  df-rel 5597  df-res 5602

This theorem is referenced by: (None)